Scottish Cup semi-finals can deliver seminal moments. Two years ago a shock Rangers victory in an Old Firm clash signalled in effect the end for Ronny Deila’s time at Celtic. The subsequent arrival of Brendan Rodgers triggered a stunning upturn in fortunes in Glasgow’s east end.

Rangers apparently lack the resources to attract a manager of Rodgers’ calibre. Nonetheless the ease with which they were swatted aside here should illustrate the scale of overhaul required at Ibrox. Had Alloa Athletic’s players moonlighted in Rangers shirts for this game, the outcome would have been no different.

Graeme Murty, who has the Rangers manager’s job only until the summer, saw his team embarrassed by a Celtic side who do not need the assistance of such lame opposition. Rangers did not have much shape to start with. They lost it all, and their discipline, long before full time. Murty, who looked shellshocked, apologised to the Rangers support as he reflected on his worst day in football.

“I don’t think we got close enough to them,” Murty said. “All four goals were avoidable. I said we had to be clinical in both boxes but I don’t think we managed to do that at all.

“I’m not here to speak about my own position; it would be churlish and selfish to do that now. I won’t talk about my role, it isn’t about me.”

In truth it is the absentee Rangers chairman, Dave King, who should be doing the talking. King continues to preside over an inefficient club which has not won a major trophy since its financial collapse of 2012.

For Rodgers, further history beckons. No Scottish team has ever won back-to-back domestic trebles. With the League Cup in the trophy cabinet and a successful top-flight defence within touching distance, victory against Motherwell in the Scottish Cup final would deliver that consecutive clean sweep.

“I think what the players have done over the past couple of seasons has inspired the supporters into believing the impossible,” the Celtic manager said. “There is always a lot of talk up here about the gap [between Celtic and Rangers] but these are tough games. Rangers have good players and there is always pressure at this club. The expectations are huge.”

Moussa Dembélé, who was to run amok, had already struck a post by the time Celtic edged ahead. Tom Rogic danced inside Ross McCrorie, with the bemused young defender left to look on as the Australian midfielder rolled a calm finish beyond Wes Foderingham.

The half-time score at least partly reflected the gulf in class. A routine Kieran Tierney cross should have been dealt with by Russell Martin. Instead the veteran centre-back played the ball straight to the feet of Callum McGregor. The return came with interest as Celtic doubled their lead. Murty’s response, which appeared little more than tokenism, was to replace Andy Halliday with Josh Windass. There were still five first half minutes to play.

Any faint hopes Rangers retained of a recovery were extinguished by a Dembélé burst. The floundering McCrorie hauled down the French striker, a red card and penalty the legitimate outcome. Dembélé took the spot-kick himself and duly made no mistake.

A Rangers flurry followed. Mikael Lustig almost scored a bizarre own goal but instead watched his sliced clearance rebound from the crossbar. Craig Gordon saved smartly, twice, from Alfredo Morelos.

The latter stages of the game were flat on the basis that Rangers had no chance of salvation and Celtic knew it. There was time for a fourth, though, again from the penalty spot, after Patrick Roberts was upended by Jason Holt. The excellent Olivier Ntcham used the opportunity to notch his eighth goal of the season.

“We have a way of playing that we know can win games,” Rodgers said. What Rangers and Murty would give for that.