Ali Crawford gave the home side a 15th-minute lead before David Templeton scored his first for the club.
Steven Boyd added a third in the 74th minute to round off a comfortable win for Martin Canning's side.
Despite a penalty shoot-out win for Queen of the South over Albion Rovers after a 2-2 draw, both sides missed out on a place in the second round.
Accies boss Canning had made eight changes from the midweek 4-4 draw with Albion Rovers.
That led to a slow start, but midfielder Crawford set the Premiership side on their way with a stunning 25-yard free-kick.
The goal settled Canning's men and visiting goalkeeper Chris Smith made further saves; one from another Crawford free-kick, twice from efforts by striker Rakish Bingham, while also denying Danny Redmond from close range.
Summer signing Templeton drilled in number two from Louis Longridge's lay-off before being replaced by Boyd, who blasted a decent chance over the bar before making amends when he beat Smith with a header.
Queen of the South twice came from behind at home to Albion Rovers in a game both sides needed to win to have a chance of taking one of the four best runners up spots.
Alan Trouten continued his fine start to the season for the visitors, scoring twice to make it seven goals in four games.
However, Stephen Dobbie and Darren Brownlie scored to take the game to penalties, with the home side earning the bonus point with a 4-2 win.
Hamilton Academical beat Stenhousemuir 3-0 to progress as winners of the Scottish League Cup's Group G.