The home side had plenty of pressure without creating too much of substance and Josh Peters almost snatched the lead for Queen’s Park with their first effort on target but Jamie MacDonald was well positioned to save.
Kris Boyd almost brought the opening goal with a header from a corner kick but the effort was just too high.
The lively Chris Burke came even closer with a superb effort from outside the area which was just inches over the crossbar.
Eamonn Brophy finally put Kilmarnock in front after 61 minutes with a six-yard shot to leave Jordan Hart with no chance.
Boyd hit the post late on and finally got on the scoresheet with an injury-time penalty.
St Mirren also got through as one of the best runners-up as they overwhelmed Dumbarton 6-0 in Paisley.
St Mirren took the lead after just two minutes, Hayden Coulson beating Grant Adam with a left-foot shot after exchanging passes with Danny Mullen.
It was a dominant performance by the home side and Mullen came close shortly afterwards but Dumbarton managed to scramble the ball to safety.
Coulson was desperately unlucky not to collect his second goal with a terrific effort midway through the first half which cracked against the post.
It was no surprise when St Mirren scored the second goal ten minutes from the break when Cammy Smith headed home a Paul McGinn cross.
Ross Stewart headed a third goal early in the second half from Mullen’s excellent cross and Stephen McGinn made it 4-0 with a great strike from outside the area. Jim Kellerman netted his first for the club before Smith completed the scoring with a wonderful strike.
Livingston won Group F with a 2-0 victory over Berwick Rangers in Linlithgow.
Player-manager Kenny Miller set up the opening goal for Jack Hamilton, who beat the goalkeeper with a low shot from a tight angle after three minutes while recent signing Egli Kaja caught the eye with some terrific skills as Livingston continued to press.
Livingston made sure of victory when Jack McMillan scored with eight minutes left.
Hamilton Academical and Airdrieonians drew 1-1 elsewhere in the group with Hamilton earning the bonus point with a 3-0 victory in the penalty shoot-out.
St Johnstone were clear winners of Group B with a come-from-behind 3-1 win at Forfar Athletic. Assistant manager John Baird put Forfar in front after great work by Dale Hilson for the first goal St Johnstone had conceded in the competition.
The Perth side equalised shortly before half-time when Tony Watt scored with a well-placed header and David McMillan put the Premiership side in front early in the second half.
Matt Kennedy made it 3-1 midway through the second half.
A double from Dylan Mackin gave Falkirk a 2-1 win at East Fife.
In Group C, Cove Rangers upset Raith Rovers 2-0 and Edinburgh City beat Stranraer 4-2 in Group G.
Michael Tidser scored a double as Morton beat Albion Rovers 5-0 in Group E and Nicky Clark also scored twice as Dundee United won 4-0 at Elgin City in Group A.