Having wrapped up the Scottish Professional Football League and the Scottish League Cup, Celtic's hopes of another clean sweep were in doubt as it 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.
"There were people saying we've had a poor season. We've done the treble," captain Scott Brown said. "It's the third one in a row and we'll look to push on again next season."
A largely uneventful first half suited Hearts and it was the underdogs who claimed the 52nd-minute breakthrough as Sean Clare teed up Australian Edwards to slam into the net after a goalmouth scramble.
But Hearts' lead lasted no more than 10 minutes as goalkeeper Zdenek Zlamal felled Edouard, who squeezed the resulting penalty into the bottom-left corner.
Edouard was again 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 Scottish champion.