The visitors last won on the road at Luton on November 5, but must have thought their luck was in after weathering a first-half storm and then sneaking ahead through Michael Ricketts just before the break.
They opted to sit on the lead though, inviting Ipswich forward and it always looked a matter of time before an equaliser which Alan Lee duly supplied on 62 minutes.
Town continued to press and won the game with an 83rd minute penalty from Matt Richards - a decision which saw Wayne Thomas sent off for dissent.
Burnley began the game well with Gifton Noel-Williams and Ricketts linking well twice, with Ricketts firing over and then having a tame shot easily saved by Shane Supple following Noel-Williams' flick.
Ian Westlake shot over as Town threatened for the first time, before another Noel-Williams flick found his strike partner who this time volleyed well wide.
Town went close on 14 minutes when Darren Currie's curling free-kick was just too high for Jason De Vos.
Then with half-time moments away, Ricketts raced on to yet another Noel-Williams flick and calmly side-footed beyond the advancing Supple.
Owen Garvan, Richards and Scott Barron all fired efforts on target as Town pressed early in the second half.
The breakthrough came on 62 minutes when Currie's left-wing cross found its way through to Lee and he used his strength to turn John Harley and fire beyond Brian Jensen from seven yards.
Town got the winner with seven minutes to go when another surging Lee run was halted by a foul from Burnley right-back Michael Duff.
Thomas was dismissed for arguing about the decision and took an age to trudge off before Richards converted his third successful penalty of the season with a shot into the left-hand corner.