Dundee United also moved into the second round thanks to a 1-0 triumph at East Fife.
Liam Boyce and 17-year-old Pollock were on target at Forthbank as Craig Gordon kept a 10th consecutive clean sheet.
After an uneventful opening half hour during which former Hearts striker Dylan Bikey drew the only save, Boyce played an inventive one-two with Gary Mackay-Steven before slotting home his third goal of the season.
Another former Hearts striker, Dale Carrick, looked set to finish after slack play in the Hearts defence early in the second half but Craig Halkett got back to put in a last-ditch challenge.
Pollock removed any doubt in the 74th minute when he converted Alex Cochrane's low cross.
Dundee United made it four wins from four games in Group B thanks to Peter Pawlett's 40th-minute effort at New Bayview. The former Aberdeen player forced a defensive mistake before converting.
Kelty Hearts joined Arbroath on six points in the hunt for a best runners-up spot as Nathan Austin's double earned a 3-1 victory at Elgin.
St Mirren moved on to nine points from three matches following a 3-1 victory at Stenhousemuir.
Curtis Main opened the scoring for the Premiership side five minutes after the break after capitalising on a defensive error, and Kristian Dennis doubled the lead midway through the half.
Thomas Orr pulled one back but Lee Erwin struck in stoppage time as Saints moved three points ahead of Dunfermline in Group H.
Brian Graham's double kept Partick Thistle in the hunt ahead of their trip to Paisley on Sunday as the Jags beat Dumbarton 2-0 at Firhill.
Kyle Connell netted a stoppage-time leveller for Kilmarnock to cancel out Lewis McGregor's effort in a 1-1 draw with Morton.
Tommy Wright's side were heading for elimination but a sudden-death shoot-out win sent Morton out and put Killie a point behind leaders Stranraer.
Clyde also earned a bonus point after a 2-2 draw with East Kilbride to put them on five points with Killie and in with a chance of topping Group G.
Ayr moved above Hamilton into top spot in Group E after Cammy Salkeld's 27th-minute strike earned a 1-0 victory in Lanarkshire.
Alloa's hopes of progress ended as Barry Ferguson's side followed up victory over Livingston with a shock 1-0 defeat by non-league Brechin.