They are now five points clear of everyone else in their pool and, after three games, have yet to concede a goal.
Kilmarnock, though, were jeered off at the final whistle by those who had bothered to stay that long. While Morton were impressive, they were desperately poor here.
Manager Lee Clark was far from pleased with his side’s showing and made no attempt to mask his disappointment.
“You can’t carry passengers when you don’t have the ball and we had too many players who weren’t doing the right thing or making a significant contribution,” he said.
“This has been a very steep learning curve for those players. It’s a reality check and we need to use it in the right way. Now we need to win the next two ties and score lots of goals to try and get through to the next round.”
The home defence was static as the Championship side knocked the ball around until Jai Quitongo engineered an opening for Gary Oliver. The former Hearts striker appeared to have blown his chance when he scuffed his shot from 18 yards, but Jamie MacDonald was still beaten low to his left as the ball went in off the post.
Clark had made eight changes to the starting XI which had beaten Clyde in their opening fixture seven days earlier and, unsurprisingly, his new-look side did not gel.
Kris Boyd was, not so long ago, one of the most feared predators in the country but, unmarked, he blazed wastefully over from 15 yards when Josh Magennis teed him up in the 11th minute.
He was frustrated again in the 33rd minute when a perfectly-timed challenge from Michael Doyle diverted his shot behind for a corner.
Magennis was inches away from the edge of the penalty area after Greg Kiltie had taken a short free-kick to the Irishman but, to the evident frustration of their supporters, Killie were creating next to nothing from open play.
Clark sent on Souleymane Coulibaly and Gary Dicker for Boyd and Mark Waddington, but the hosts remained bereft of vigour or vision and they deservedly fell further behind at the three-quarter mark.
Jonathan Burn, the centre-back making his debut on loan from Premier League Middlesbrough, was at fault when he was muscled off the ball by Quitongo.
The defender then resembled a man towing a caravan as he attempted to catch the 18-year-old, who comfortably outpaced him before MacDonald, also culpable, was beaten by a low drive almost from the touchline.
Quitongo marked his first-ever goal with an acrobatic backwards somersault and almost added to the embarrassment in the 85th minute but, after waltzing through the porous Killie defence, he drove the ball straight at the advancing goalkeeper. However, that could not detract from what had been a momentous afternoon for him.
“I have dreamed of fans signing my name since I was a kid,” Quitongo – son of former Hearts and Hamilton winger Jose – said. “It was extra special because I had a bet with Tam O’Ware. He said I wouldn’t score all season and now he has to take me to any restaurant I want.
“I had planned my celebration. I have done it since I was a wee boy. My dad used to do the cartwheels, but I think my celebration was better.”