Scott Parker's side was struggling until Edwards handed them the lead just before half-time.
With the Championship's top two Leeds and Sheffield United playing each other on Monday, this was a chance to close the gap on the Yorkshire pair and Burnley took advantage with second-half goals from Josh Brownhill, Connor Roberts and Manuel Benson.
Edwards, signed on loan from Sporting Lisbon last month and who was in the Tottenham's academy when manager Parker was there, opened the scoring after 43 minutes.
He had already clipped the crossbar with a deflected 20th-minute cross that represented Burnley's only threat to the visitors' goal until he struck.
Having played a short pass to Josh Cullen, Edwards received the return, benefitted from a ricochet off Shea Charles and showed neat footwork before producing a close-range finish.
It was a cruel way for Danny Rohl's team to fall behind after they had made light of Burnley's incredible defensive record this season. The opening 34 league games have seen the Clarets concede just nine goals.
Wednesday missed three great chances to end Burnley's run of 11 games and 1,000 minutes without conceding in the first 19 minutes alone.
Michael Ihiekwe produced a shot which clipped the outside of the post, winger Djeidi Gassama crossed for Stuart Armstrong, who steered a six-yard shot wide, before Gassama was again the architect, this time setting up Josh Windass, whose effort flew wide via defender Maxime Esteve.
Wednesday started the second half as they had the first, with Windass and Armstrong both having glorious chances blocked by defenders.
As had been the case earlier, Wednesday paid for those misses with Brownhill scoring with his first touch after coming on as a 62nd-minute substitute.
Burnley maintained the pressure from a half-cleared corner, with Josh Laurent sending Jaidon Anthony away down the right and his cross being met by Brownhill, who scored from close range after a superbly-timed run.
The two-goal cushion allowed Burnley to play with more composure and Roberts was somehow unmarked to add a third after 70 minutes, side-footing in from Anthony's cross.
"Full credit to everyone at the club, for us to be able to get a player like Marcus," said Parker, who watched the game from the stands while serving a touchline ban.
"I understand his quality. The first time I experienced that was as a 13-year-old at Tottenham and now he's a Champions League player and he showed his quality throughout."
29°C