Charlie Austin headed an early chance over for the Championship outfit and winger Junior Stanislas curled a free-kick narrowly wide.
Bury's best two chances of the first-half both came from David Worrall crosses, but skipper Steven Schumacher and Shaun Harrad could not convert.
Schumacher should have put the League One side in front as the hour approached when Worrall's right-wing corner was flicked on at the near post and he failed to find a decisive touch with the goal gaping.
Burnley boss Eddie Howe sent out a fully changed XI after the break, with Keith Treacy and Martin Paterson linking well in attack. The latter almost turned home his team-mate's cross moments after Schumacher's miss and Treacy went on to sting the palms of Bury goalkeeper Trevor Carson with a rasping drive.
But it was Ings, the third member of the Clarets three-pronged forward line, who settled the contest nine minutes from time, racing onto Treacy's through ball to coolly clip past the advancing Carson.