With just 12 minutes remaining, Welch picked up a loose ball just inside the Hatters half, before going on to unleash a dream shot which rifled into the top corner.
Stockport opened brightly in their second home game of the season. Glenn Murray side-footed a shot wide inside the first minute, then Adam Griffin headed a pinpoint cross from Keith Briggs disappointingly over the crossbar.
On-loan striker Murray was full of running early on, and on nine minutes he latched on to a deflected shot from Griffin, but could only poke the ball narrowly wide.
He didn't have to wait too long to get his reward, however. Just three minutes later Murray produced a sweet finish from 15 yards after Griffin had sent in another tempting cross.
Goalscorer Murray almost repeated his effort shortly after when he turned a low shot just past the post, following a Rob Clare header into the Stanley danger zone.
Romauld Boco, last weekend's Stanley goalscoring hero, saw a deflected shot comfortably saved by James Spencer, then on 29 minutes Leam Richardson crossed for Andy Todd, who glanced a header inches wide.
Stockport remained on top as half-time approached and next to go close was defender Gareth Owen, who saw a goalbound header blocked five yards from goal.
Griffin followed that up with a crisp volley which flew just past a Stanley post, but the visitors hit back when Gary Roberts curled a 25-yard free-kick inches over Spencer's crossbar.
Accrington's Ian Craney had the final word of the first half when he twisted and turned before unleashing a 20-yard drive which curled inches wide of the post.
Murray caught the first sight of goal after the break when he produced a neat turn before seeing a low drive deflected over for a corner.
Stanley replied on the hour mark when Craney's curling free-kick was deflected inches over the crossbar, then Spencer produced a stunning finger-tip save to keep out Todd's flicked header.
Spencer then kept out a thumping Boco header with a similarly impressive flying save.
The visitors ended the game on top and even after Welch's thunderbolt, substitute Sean Doherty should have won it when he stroked a simple shot wide from 15 yards, with Spencer for once stranded.