The club entered administration in June with debts of nearly £30m, with Lithuanian firms Ukio Bankas and UBIG, who own 80 per cent of the shares, due the bulk of the money.
UBIG abstained from the Friday's vote but the vast majority of those taking part backed a £2.5m rescue bid being offered by fans group Foundation of Hearts.
Ian Murray MP, chairman of FoH, said: "Today I'd like to thank our supporters - without them it would not have been possible to get this far.
"Whilst the journey is not over yet and there is much work to be done, we can move forward with a degree of optimism that Heart of Midlothian can have a future we can be very proud of.
"It is because of you, our supporters that we have achieved this significant step, but bigger challenges lie ahead and there is much to do. We must remain united with purpose."
UBIG's 50% holding cannot be sold or assigned until they have their own creditors' meeting, which is scheduled for late January, so any exit from administration could not happen until February at the earliest.
Gintaras Adomonis, bankruptcy administrator of Ukio Bankas, said: "I should admit the fact that our decision has a substantial part of emotion in it. We are impressed by the efforts the fans of Hearts FC have put in the process and we were convinced that the CVA proposal was the best way to save the club.
"Therefore Ukio Bankas voted for approval and will be looking forward to implementing the CVA if all the outstanding issues are sorted and positions clarified."
Hearts entered administration in June and started this season with a 15-point deduction in the Scottish Premiership.