Having wrapped up the Scottish Premiership and the Scottish League Cup, Celtic's hopes of another clean sweep were in doubt as they trailed early in the second half to Ryan Edwards' strike.
But club-record signing Edouard stepped up with an equaliser from the penalty spot and a late second to clinch the treble-treble for the first time in Celtic's history.
"It's phenomenal. For the fans, it's unbelievable and that was people saying we've had a poor season. We've done the treble yet again and it's the third one in a row," captain Scott Brown told BBC Sport.
"It shows you the lads are relentlessness. They threw the kitchen sink at us and we held out. Hearts have given us a very hard, very physical game, but we stood up to it.
"We know, if we turn up, we win the game."
A largely uneventful first half suited Hearts and it was the underdogs who claimed the 52nd-minute breakthrough as Sean Clare teed up Edwards to slam into the net following a goalmouth scramble.
But Hearts' lead only lasted 10 minutes as goalkeeper Zdenek Zlamal felled Edouard, who squeezed the resulting penalty into the bottom-left corner.
And Edouard was allowed through a second time late on to net the winner, keeping his cool and lifting a finish over Zlamal to cap another sensational season for the champions.
Hoops boss Neil Lennon was quick to hail his side afterwards and the magnitude of what they have achieved.
He said: "You'll never see this again. Edouard's a superstar.
"In footballing parlance, it's impossible [to put it into words]. That's nine trophies out of nine. It is remarkable.
"They've done themselves great justice and they've done the club proud."
Quizzed on his own future, with a short-term deal coming to an end, Lennon said it was "too early" to say what will happen.