Duffy was looked upon as a marquee addition when the Hoops tempted the Republic of Ireland international on loan for the 2020/21 season. However, the player’s form mirrored that of his club as Celtic suffered a disastrous campaign as they chased ten-in-a-row.
Manager Graham Potter insisted the 29-year-old would bounce back from his disappointing move to his boyhood heroes and was confident enough to name him from the start for the season opener.
Burnley took the lead via a second-minute James Tarkowski header, which stood after a VAR check amid Brighton claims that the defender had fouled Maupay, and the Clarets went on to rattle the woodwork twice before half-time, Johann Berg Gudmundsson hitting the post and Ben Mee the bar.
The visitors subsequently capitalised on those near-misses after the break as two substitutes helped them turn things around shortly after entering the fray, Jakub Moder teeing up Maupay's equaliser in the 73rd minute and Mac Allister then netting the winner five minutes later.
Burnley fans were cheering within moments of the contest getting under way after Tarkowski sent Ashley Westwood's corner past Robert Sanchez.
Brighton protested that Tarkowski had fouled Maupay before heading in, the pair having collided as the Burnley defender moved towards the goal and Maupay ended up on the turf, but the effort was allowed to stand following a VAR check.
The hosts subsequently picked up two bookings in quick succession, with yellow cards being shown to Gudmundsson for a challenge on Yves Bissouma, and Tarkowski after another coming-together with Maupay, which resulted in a free-kick that Solly March sent into Nick Pope's arms.
Gudmundsson then almost made it 2-0 in the 19th minute when his low strike come back off the post.
Another Tarkowski-Maupay challenge led to a booking for the Brighton striker, before Jay Rodriguez's shot brought a save out of Sanchez.
Burnley then went close to doubling their lead again in the 36th minute as skipper Mee headed against the bar and the ball bounced down on the line, with Tarkowski firing over on the follow-up.
At the other end, Duffy headed over and Pope had little trouble catching an acrobatic attempt by Maupay.
Rodriguez going down in the box under the attentions of Duffy sparked penalty calls from Burnley that were waved away by referee David Coote, and Leandro Trossard then lofted an attempt wide of Pope's goal just prior to the hour mark.
After Chris Wood saw a shot deflect behind, Brighton boss Graham Potter made a change by bringing on Moder for Steven Alzate - and the substitute had a near immediate impact, sending a cross in from the right that the sliding Maupay converted to draw things level.
Moments later, Potter then saw another substitution swiftly pay off as his side grabbed the lead, with Mac Allister, sent on for Trossard soon after the equaliser, slotting past Pope from Pascal Gross' delivery.
Burnley subsequently pushed for a leveller in vain, while a late Gross attempt looked like it could be heading in before Tarkowski made the clearance.