It was not until 19 minutes to go that former Scotland international Fletcher slammed the hosts in front from the penalty spot after Smith had been felled by Brewers goalkeeper Max Crocombe.
But a cool-finish from Smith three minutes later removed any lingering worries, with Marriott adding to the jubilant atmosphere when he provided a third on 87 minutes.
Burton, though, had asked serious questions of Wrexham as they defended superbly despite their early setback.
The Brewers will take heart in their fight at the other end of the table as they remain in the bottom three with six games remaining.
Wrexham have five games to go in their attempt to make history with a third successive promotion. They head to Wigan next weekend while Wycombe play their game in hand at Huddersfield on Tuesday.
Wrexham supporters, who seemed to acknowledge the potential importance of the afternoon’s events with the way they celebrated at the final whistle, will hope not all games are as tense.
Indeed, they had to show similar patience to their side after Webster’s early sending off gave Burton enough reason to put up a yellow wall of resistance.
Webster walked off shaking his head after seeing the red card, although replays suggested referee Seb Stockbridge made the right call after the midfielder caught Dobson with studs showing.
Bowyer displayed his frustration, his team responded with determination.
Sat deep, they kept Wrexham at bay and waited for their own moments. Had Wrexham goalkeeper Arthur Okonkwo not been alert, Julian Larsson may well have broken the deadlock as he raced behind after a set-piece was played back in on 35 minutes.
The introduction of Elliott Lee and Ryan Barnett boosted the hosts’ the urgency after the interval, but it was not until Smith raced onto a hopeful through ball that the game swung, Crocombe racing off his line and the striker welcoming the contact after getting the crucial first touch.
Fletcher, on for Jay Rodriguez who suffered a first-half injury, made no mistake from the spot with his eighth of the season before Smith doubled the advantage when he slotted home expertly from Ryan Longman’s through pass.
And Marriott, another off the bench, wrapped up things when he finished after breaking through late on.