Rangers' slip-up on Saturday against Aberdeen meant Neil Lennon's champions could move seven points clear of their stuttering rivals with victory at the Foys Stadium.
Yet they found themselves behind after Marios Ogkmpoe fired Accies ahead midway through the first half.
But Brian Rice – serving the first of his five-game ban for breaching betting rules – saw his side's chances slip away when Jamie Hamilton was dismissed for a last-man foul on Leigh Griffiths.
Edouard fired home the resulting set-piece and added a late second three minutes after Christopher Jullien had put the Hoops back in charge of the game and the title fight.
James Forrest completed the Parkhead side's late charge to leave Celtic with their sights firmly set on winning a ninth straight title.
However, Accies, are now staring at a fight for survival after slumping to the bottom of the league.
But David Templeton started on the bench 48 hours on from sealing his Hamilton return.
The motivation was clear for Lennon's team after Gers' dropped points for the second time in a week.
Yet they found themselves in an arm wrestle with the hosts, who were able to wriggle free time and again from pressure as they found one of their two strikers up field.
Griffiths got himself involved in awkward tangle after 13 minutes as he pulled Sam Woods down before planting his foot into the defender's prone body.
Referee Nick Walsh gave the striker the benefit of the doubt as he flashed only yellow, while also booking Alex Gogic for his angry reaction as both sets of players stormed in for a squabble.
But there was a sense of complacency about Celtic.
Kris Ajer got his header all wrong as he went to meet Aaron McGowan's cross. The ball fell for Ogkmpoe but Fraser Forster beat away his volley.
Jullien then fired the ball out of play under no pressure while Forrest was surprisingly burned for pace by Scott McMann as Hamilton won a corner.
Jullien then failed to get a grip on Ogkmpoe as Blair Alston put in the delivery, allowing the Greek to power home with his head after 27 minutes.
But the game swung back in Celtic's favour 10 minutes before the break.
Jamie Hamilton got caught the wrong side of Griffiths as they chased a Jullien long ball, hauling the Hoops frontman down right on the edge of his own box.
Walsh had no option but to flash red at the Accies youngster, with bad swiftly becoming worse for Accies as Edouard swept the free-kick over the wall into the corner of Luke Southwood's goal.
Celtic sensed blood and went for the jugular after the half-time turnaround. Griffiths blazed one chance over before smashing his next set-piece opportunity off Southwood's bar, while Woods and McGowan were both relieved to survive handball shouts in the box.
Ryan Christie made his return from groin surgery just past the hour mark while Accies swiftly handed Templeton his second debut.
But the hosts had to resort to some desperate last-ditch defending as their box came under siege.
Hamilton's resistance could only last so long and Celtic pounced for the win with 12 minutes left.
Jonny Hayes slipped fellow substitute Christie down the left and his cross was steered home by Jullien's expert finish.
And any doubt about the result extinguished three minutes later as Tom Rogic charged forward before slipping in Edouard for his 22nd goal of the season.
Forrest completed the scoring in stoppage time as he combined with Edouard before rifling home the fourth.