Oan Djorkaeff scored a late penalty to spare St Mirren consecutive League Cup upsets and earn Jim Goodwin his first win as the Paisley club's manager.
The Premiership side, who lost to Dunfermline Athletic in their opener, were frustrated by League Two Edinburgh City until Paul McGinn was brought down and Djorkaeff - son of French World Cup winner Youri - netted from the spot.
Top-flight rivals Hamilton Academical recovered from losing in a shootout to bottom tier Queen's Park on Saturday by beating League One Clyde 3-1.
There were also comfortable wins for Championship sides Dunfermline Athletic and Arbroath, while second-tier rivals Dundee were held to a goalless draw by SPFL newcomers Cove Rangers before claiming a bonus point for winning the shootout.
The eight group winners and the four best runners-up all advance to join Scotland's four European representatives in the last 16.
Dundee earned a bonus point for beating Cove Rangers on penalties
After failing to win their opening matches, the pressure was on both St Mirren and Hamilton to revive their hopes in the competition.
With Dunfermline romping to victory against Albion Rovers, the Paisley side were poised to slip five points behind the Fifers with two games remaining and effectively out of contention to reach the last 16. That was until Craig Thomson fouled McGinn in the 88th minute to allow Djorkaeff the chance to score his first goal since signing last week.
It was harsh on League Two Edinburgh City, who looked like taking a point from their first game in the competition this term, but priceless for Goodwin, whose side move up to second place. "I was really pleased with the players in terms of how they kept going," said the manager. "I can't fault the lads. Their attitude is great."
The pressure on Hamilton counterpart Brian Rice was not as extreme, not least because the Lanarkshire side have habitually struggled in the group stages.
Regardless, Scott McMann's goal just after the half hour settled any nerves and Blair Alston doubled the advantage just after the break. Kristoffer Syvertsen pulled one back, but Lewis Smith put the result beyond doubt to move his side to within two points of group leaders Partick Thistle.
"It was a difficult game on a difficult surface but we played really well," said head coach Rice.
Dundee could not find a way past League Two newcomers Cove
Cove Rangers have insisted that their aim is to simply stay in League Two the term after their promotion into the SPFL, but the Aberdeen club offered ample evidence of their qualities by holding Dundee.
Under the stewardship of former Dens Park manager Paul Hartley, Cove frustrated aside relegated from the top flight in May and earned a point. Losing 3-2 in the shoot-out only took some of the sheen from the performance.
Dundee, meanwhile, edge above last season's League Two champions Peterhead on goal difference to top the only section not to contain a Premiership team.
The third round of matches begins on Friday, with Premiership side Motherwell hosting second-tier outfit Morton, while Championship club Dundee United welcome League Two Cowdenbeath. A further 12 games follow on Saturday, with the final two fixtures of the set the next day.