Burgess had a miserable campaign last season ruined by injury and poor form.
But his clinical strike after clever work from Andy Morrell gave Blackpool their 12th successive victory.
What started slowly swiftly became a full-bodied cup tie and Huddersfield forward Chris Brandon will be regretting failing to make more of a clear-cut chance with five minutes to go when ex-Terriers keeper Paul Rachubka dived bravely at his feet.
Both sides had first-half chances with Blackpool's Keigan Parker twice going close.
Michael Collins had an even better opportunity on 34 minutes but planted his shot straight at Rachubka.
Both sides played simple, direct football but, with no-one up front to give Blackpool some aerial power, the Huddersfield back four had an easy time of it until Burgess came off the bench.
With him came midfielder Wes Hoolahan and his control of midfield saw Blackpool step up the tempo and dominate the exchanges.
Morrell and Gary Taylor-Fletcher both came close and with a minute to go David Fox clipped the woodwork with a glorious 40-yarder.
Seasiders boss Grayson was left with one headache. Defender Ian Evatt came off with two minutes of the first half remaining with what looked like a pulled hamstring.
Latvia international Kaspars Gorkss replaced him and he too suffered an injury in a heavy tackle from Malvin Kamara.
The tie also took its toll on Andy Ritchie's side with the experienced Frank Sinclair being replaced with 20 minutes to go after suffering a knee injury.