The fabled ‘new manager bounce’ is one thing. Stuttgart winning away, though? That’s something else entirely and after a first victory on the road since winning promotion back to the top flight, Tayfun Korkut is earning the right to be taken seriously.

As the final whistle went on Sunday’s victory at overachieving Augsburg, there was a genuine sense of achievement. Stuttgart’s players celebrated not so much, it seemed, for getting one over near neighbours in Swabia but for the sensation that maybe a corner really has been turned. It has long been received wisdom that only maintaining their strong home form would keep Stuttgart up. That’s a fair theory on which to base a campaign for survival but giving up hope of ever getting anything from away games was somewhat less so.

Now, less than a month after Korkut was the deeply underwhelming replacement for the deposed – and very popular – young coach Hannes Wolf, it has begun to seem a brighter future is possible. Wolf lost six of his last seven games in the Bundesliga and Stuttgart failed to score in five of those, so Korkut was inheriting an unenviable situation. When he took the helm, they were four points ahead of the bottom two and three clear of the relegation play-off spot.

Now, after a pleasantly surprising start of seven points in three games, Die Roten are four points north of the play-off place but – crucially – have a 10-point cushion between themselves and second-last Hamburg. The sporting director, Michael Reschke, warned: “We need to keep the calculator” amid the celebrations on Sunday. The new man hasn’t forgotten where he, or his team, have come from, either. “We should be careful about getting too euphoric,” Korkut cautioned. “There are still a few stones on the path we have to push out of the way.”

That is exactly what Korkut expected at the beginning, certainly. He must interview really well, people assumed, after he was appointed on 29 January to much bemusement. Some of that came from the decision to dispense with Wolf, an impressive man who – as previously documented in this column – Borussia Dortmund were very sad to see leave their ranks, where he had flourished with the under-19s. Korkut, on the other hand, was last seen being ushered out of the back door at Leverkusen after an unexpected brush with relegation trouble, having previously failed to deliver at Hannover and Kaiserslautern.

So how was he going to help Stuttgart? He was neither a name nor a survival specialist in the mould of, say, Bruno Labbadia. He had the advantage of being free and that was about it, with the prevailing mood that Wolf had been harshly treated adding to the cynicism. Now, however, Stuttgarter Zeitung write about the new coach “astounding [us] with his tactical tricks and special match plans”.

The difference on the field is undeniable. That Stuttgart now play 4-4-2 might not seem that groundbreaking but going with two looming forwards in Mario Gómez and Daniel Ginczek has been the coach’s attempt to put pressure on the opposition. It’s worked, albeit without making the team free-scoring, with three goals yielding the seven points under Korkut. They made chances but without finishing them. “We could have won 3 or 4-0,” Gómez lamented.

There has been other creative thinking, such as using Holger Badstuber as a central midfielder on his return. It’s a position Badstuber couldn’t recall playing since his days in the Bayern reserve team a decade ago but he and Santiago Ascacibar acted as an excellent shield for the defence. “If that’s where I’m needed, that’s where I’ll play,” Badstuber said.

Stuttgart’s Santiago Ascacibar challenges for the ball with Koo Ja-cheol. Photograph: Matthias Balk/AP

It’s not pretty, of course. “Football-wise, some teams are certainly better than we are,” Gómez added. “Our biggest asset is that everyone pulls together.” The 32-year-old, who had already spoken of feeling renewed by returning to his first club, came back from missing training at the end of the week with flu to put in a good shift, though no moment was more important than his winner. He expertly brought Erik Thommy’s blocked free-kick under control before sweeping it into the corner past Marwin Hitz.

Gómez deserves credit for his influence on the group, too; a senior player who really befits the description, he has stepped up to galvanise the squad on Korkut’s arrival, just as he did for Wolfsburg last season. Those goals made all the difference for Die Wölfe. While Stuttgart might be grateful of the boost, they’ll be hoping it’s not nearly as much of a close escape as it was for Gómez’s old team.

He is hoping for a little more from the end of the campaign, telling Welt am Sonntag in an interview about his renewed hopes of making Germany’s World Cup squad – and seemingly putting Sandro Wagner, who spends much of his time describing himself as Germany’s greatest centre-forward, in his place. “[Joachim Löw] knows he can count on me,” Gómez said. “Of course, if I was at Bayern now, you would expect a few more goals. If you don’t score there, you would have to worry.”

Any call-up for Gómez would be greatly celebrated in Stuttgart but there is still work to do. While the plan to contain Augsburg worked a treat, there were still a few moments underlining the fragility of their success. VAR – correctly – ruled out a potential equaliser for Michael Gregoritsch in the first half and a few hearts were in mouths as the home side appealed furiously for a penalty in stoppage time when Gregoritsch’s header hit Badstuber’s hand, which the referee, Tobias Stieler, decided was from too close a range to be deliberate.

Still, they are in much better shape than they were, steeled for the tasks ahead and, as Stuttgarter Zeitung put it: “Guided by a coach who has been able to turn the great scepticism into appreciation and trust.” Few would have expected that to be the case three weeks ago.

Talking points

• The aforementioned Wagner won’t be stepping aside without a fight, of course. On a rare start before Bayern’s upcoming Champions League commitments, he headed a valuable equaliser at Wolfsburg, a game in which his side had trailed for the best part of an hour and missed a penalty via Arjen Robben. Robert Lewandowski came on to snatch a win with a stoppage-time penalty.

Robert Lewandowski celebrates scoring Bayern’s second goal. Photograph: Fabian Bimmer/Reuters

• Marco Reus, Mario Götze and André Schürrle, remarkably, started for the first time together with Borussia Dortmund at Borussia Mönchengladbach, and the trio combined for Reus’ first goal in 274 days, a superb dipping shot that won the game. Booed by home fans on his return – which he said “hurt a lot” – Reus was consoled by the Gladbach sporting director, Max Eberl.

• Leipzig have the chance to take back second from Dortmund by winning at Eintracht Frankfurt on Monday night, although a victory would take the home side third. Schalke are back in contention after a much-needed win over Hoffenheim, courtesy of Breel Embolo’s winner, gift-wrapped by Kevin Vogt.