Hatters disappeared from the Football League in 2009 under penalty of 30 points but the appointment of John Still in 2013 seems to have finally done the trick

Kenilworth Road, Tuesday: an evening when cold threatens to take residency in cartilage. On Luton Town's reception desk, Lita Nunn is fielding calls and reassuring questioners: tonight's Skrill Premier match against Wrexham is not a sell-out – yet.

"I joined the club in 1985, during the glory days," she says. "I used to give players their wages in brown packets. I thought Emeka Nwajiobi was joking the first time he said his name." The club was such a family back then, she adds, that the backroom staff had dinner at the Savoy with players after Luton's greatest moment, the 3-2 Littlewoods Cup victory over Arsenal in 1988.

And now? The old vibe is returning, she says. Everyone is happy. And with good reason: the club are on the up – and going up.

Wrexham's players arrive. One carries a cardboard crate of Lucozade. Another a beatbox that has seen better decades. Their caretaker manager, Billy Barr, wishes everyone well. Later he will have little to smile about. From Luton's first attack, Luke Guttridge steers a shot into the bottom corner before uppercutting the sky as the crowd of 7,526, Luton's third biggest of the season, shout: "Easy! Easy!" Then the striker Andre Gray digs the ball out of a sandy pothole, scampers into the penalty area and scores. Two-nil.

The match is eight minutes and five seconds old. And over.

It ends 5-0. Luton's last league defeat, 2-0 at Wrexham in September, has been emphatically avenged. The Hatters are now unbeaten in 24 matches and 14 points clear of their nearest rivals in the Conference, Cambridge United. They have scored 17 unanswered goals in three games, taking their goal difference to +55. But it is the manner of their play – direct but not lumpen, via ground rather than air, intricate despite the sticky surface – that impresses as much as the score.

At the final whistle the corrugated iron in the main stand roof is pounded by boot and fist. It is as if the West End production of Stomp has been transported north. The supporters are boisterous, giddy: "That's why we are the champions," they chant.

They are not there yet. Not officially. But it is only a matter of time. Forget the cold: the fans can feel something else in their bones. The tingle of their club emerging from the woes and blows of recent years.

Afterwards, Wrexham's caretaker manager gives a glowing assessment. "Luton are the best team I have seen in the Conference for many, many years," he says, before telling the Guardian that he expects Luton to go up next season too.

Luton's managing director, Gary Sweet, is aiming even higher. "If you look at Swansea, Reading and Hull, I don't think any of them have the football heritage and catchment area that Luton has," he says. "Our time will come. We have always believed that. We will get promoted. And we'll get promoted again. And then we'll get promoted again."

But this is Luton we are talking about. There is an early episode of The Simpsons in which Homer attempts to leap over a canyon on a skateboard, falls short, and thumps his head on every jagged edge on the way down. And then, after being airlifted to an ambulance he falls out of the back and down the cliff again. Over the past 15 years Luton have undergone what seems like a similar experience.

In 2006-07 they were in the Championship play-off positions after 13 games, before suffering three successive relegations. In the past 15 years they have been through three administrations, 40 points worth of deductions, faced 55 charges from the FA for systematic abuse of transfer regulations and have owed the taxman £3.5m. Then there was John Gurney who, having bought the club for £4 in 2003, tried to rename it London Luton and introduced a "manager idol" phone vote for a new boss.

"But the lowest point was coming out of the FA offices in August 2008 knowing that we had a 30-point deduction and that we were going to get relegated to the Conference the following May," Sweet says. "It knocked the stuffing out of us. And this was shortly after we paid a few million to buy the club just as the global economy was entering recession."

True, there has been the occasional giddy high – particularly last year's FA Cup fourth-round victory at Norwich City and a Johnstone's Paint Trophy success in 2009 – but what was expected to be briefest of sojourns in the Conference has lasted five years. Three times they have missed out in the play-offs.

A year ago this week, with the club facing another season in non-league football, Luton dispensed with Paul Buckle and brought in John Still from Dagenham & Redbridge. Still spent the last few months of the 2012-13 season assessing his squad and then got rid of a dozen of them.

He brought in old hands, such as the 34-year-old Paul Benson, and fresh blood, such as the 20-year-old defensive midfielder Pelly Ruddock from West Ham United and Norwich's FA Youth Cup-winning captain, Cameron McGeehan, on loan. There was no more deadwood or driftwood. Still had a squad he could chisel and polish.

"He brought the right players in, not just those who want to come in for a payday, and he never takes the foot off the gas," says Gray, the club's leading scorer. "He has got us together as a team more. We're fitter than ever before. He drills us very well. Pass and move, and it's evident in our games."

Still, meanwhile, admits he wasn't sure whether to join Luton. "When they approached me I spoke to some people about coming and they advised me not to," he admits. "But there is a massive expectation here and that gave me a buzz."

Some Luton fans believed they deserved promotion because of their size. "But there is no divine right," says Still. "It's about hard work. Discipline. Pattern of play. But the biggest thing we've done we've involved everyone in the team. We're a team now, on and off the pitch. That is massive to what we have done."

Change didn't occur overnight. Luton lost their first game of the season, away at Southport, and bobbled along for the opening two months. But, according to the club chairman, Nick Owen, there was a pivotal moment in late September when a fan, who had been shouting abuse when the team were 1-0 down against Lincoln, was confronted by the captain, Ronnie Henry. "After the game, which Luton won 3-2, Ronnie said: 'This can't go on: we are all doing our best,'" explains Owen. "The fan came to see the players in training. And it broke a spell of antagonism that had built up during all these years of discontent. Now everyone is onside."

As if to prove Owen's point a couple of weeks later, Luton came from 3‑1 down to beat Halifax 4‑3 and move up to fifth. After every home game, when the players huddle together on the pitch and Still dispenses a few words of wisdom, a few lucky fans are invited to join in, emphasising the bond between player and watcher.

On the entrance to the club shop there are images of the plastic fantastic team of the 80s, including Steve Foster, Brian Stein, Danny Wilson and Ricky Hill. Stein, who scored the winning goal at Wembley when Luton won the League Cup, is liking what he sees from the new generation. "Something is happening at this club," he says. "They lost the spirit for a bit and the place was in turmoil for a while. But things are turning around."

In the longer term, Luton have plans to move to a new stadium – although, as Owen ruefully recalls, even as far back as 1958 the Evening Standard was announcing that Luton were planning the country's finest new ground – and 56 years on, after many schemes and dreams, the club are still stuck in Kenilworth Road, that footballing Tardis, squeezed between rows of terraces houses.

Yet Owen is not too despondent. "I know people say it's one of the worst grounds in the country but for me it is a very special place, full of soul and history and emotions," he says. "And after so many tough times it is great to see it energised again."