The visitors will feel hard done by after a number of key decisions went against them, including the one to award Brighton the penalty that saw them open the scoring in the 35th minute.
A Brighton corner resulted in the ball being headed onto the arm of Leicester defender Jannik Vestergaard and the spot kick was confirmed by a VAR review despite the replay showing he was being fouled by Neal Maupay.
Maupay blasted home the penalty as irate Leicester players continued to remonstrate with referee Stuart Attwell.
Welbeck made it two five minutes after the break, stealing in behind Vardy to score with a glancing header from a free kick, but Vardy was not ready to let the game go.
The striker pulled one back for Leicester in the 61st with a close-range finish after a delightful ball from Youri Tielemans and the visitors surged forward looking for an equaliser.
The Foxes had the ball in the net twice more before the final whistle but both times Harvey Barnes was judged to be in an offside position as the home side hung on to win.
The victory lifts Brighton to provisional third spot on 12 points, one behind leaders Liverpool and Manchester United, though they could be overtaken if Chelsea win at Tottenham Hotspur later on Sunday. Leicester are 12th on six points.