The fans' boss leading their rescue package has now warned they face a battle against the clock to survive.
The Jambos secured the backing of their creditors and shareholders on Friday morning for a Company Voluntary Arrangement which could eventually see supporter group Foundation of Hearts take control of Tynecastle as part of a £2.5million deal.
The double vote is the first two legs of a five-stage process that could run until early February before it can be completed.
However, administrators BDO have warned that they only have enough funds to see the club a few weeks beyond that time scale.
The key now lies with Lithuanian firm UBIG, who own 50 per cent of the club but are themselves in the midst of insolvency action.
They have yet to appoint their own creditors' committee to rule on whether to hand over their shares to FoH, and that might not happen until after the new year.
However, FoH chairman Ian Murray admitted the club have to complete the process as soon as possible.
He said: "There is still a long, long way to go and we do need to manage expectations, but it is a great day for the club.
"First and foremost, though, the priority must be to keep the club financially stable until the process runs through.
"BDO have already stated that they have enough resources to keep going until the end of February so we have to look at that now as the back-stop date."
Gary Locke's side face Celtic on Sunday in the fourth round of the William Hill Scottish Cup and Murray said a result against the Hoops could buy the club some more time.
"Sunday is a huge, huge game and a draw or a win will help swell that pot of resources the club has to work from," he said.
Hearts went into administration in June owing almost £30million to UBIG and Ukio Bankas - both once controlled by former club owner Vladimir Romanov.
Like UBIG, Ukio Bankas are major shareholders, owning around 29.5 per cent of the Jambos as well as a floating charge on Tynecastle but are in administration too.
But before the club can exit administration, they must now secure UBIG's stake, then agree a sale and purchase agreement with the firm's administrators before completing the whole deal before the money runs out.
Trouble-shooter Bryan Jackson of BDO said: "This is a very important day for Hearts - if we had not got this CVA agreed, there would have been no going forward.
"We hadn't considered a 'Plan B' because there wasn't one.
"We really had to get today's obstacles out of the way and get it voted through.
"I hate being asked to quantify the club's survival chances but they are better today than they were yesterday. It's a step in the right direction.
"We still have to deliver the 50 per cent (UBIG) shares and the sale and purchase agreement are the other hurdles to be cleared. In Lithuania, the UBIG administrators need to convince their court and creditors that this is the best show in town, so it's not done yet."
While Ukio Bankas gave their backing for the CVA, UBIG abstained from both of Friday's vote which saw 75 per cent of the creditors and all of the shareholders taking part, cast in favour of the rescue deal.
But Jackson explained UBIG refraining does not mean Hearts are out of the woods just yet.
He said: "I'm actually very grateful for UBIG and their abstention because the difficulty we have in this situation is the fact that the legislation is different in Lithuania. There are impending changes too which make it even more complicated.
"(The administrators of UBIG) are responsible to their creditors and their court procedures but their appointment was very recent. We are asking them to vote on a document like this very early on.
"They wanted to postpone again - there was nothing sinister in that - but we persuaded them to let us go ahead.
"We explained they would not prejudice their position if we went ahead with the CVA. Whilst we have that part of it voted through, the whole things falls if we can't agree the sale and purchase agreement and get the 50 per cent shareholding. So they agreed to abstain.
"I would hope, and it all depends on Lithuania, that we would be quite advanced by February. Aspirations that we will have this done by January are not realistic."
More than 7,500 Hearts fans have pledged cash to the FoH to save their club and the Ukio Bankas administrator Gintaras Adomonis said the hard work of the supporters had swayed his vote.
He 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 have put into the process and we were convinced that the CVA proposal was the best way to save the club."