That is despite the result being an eight-month delay in Hearts passing into fan ownership.
Present owner Ann Budge had proposed delaying the repayment of her £2.5m investment to help fund the project.
Following a vote of foundation members, only 43 were against the proposal and 3,790 in favour.
Hearts said on their website: "We are absolutely delighted that the foundation members have given such a resounding vote in favour of the offer from Ann Budge.
"This significant milestone on the road to the 'new' Tynecastle once more demonstrates our collective strength."
Hearts hope to fund the new stand via £3m from club reserves, £2.5m from benefactor contributions, £0.5m from commercial contributions, which leaves a shortfall of £6m once the club adds a contingency for an increase in costs.
The Scottish Premiership club hope to build up further reserves and launch a new debenture scheme, but the delay of up to two years in the Foundation of Hearts buying Budge's stake will allow funds to go directly into the club in the meantime.
"Assuming contributions continued at today's levels, majority ownership would pass to FOH only eight months later than originally envisaged," Hearts have stated.
Full planning permission is still required from City of Edinburgh Council, but Hearts have already planned out the progress of the two-year project that will result in a stadium with a capacity up to 21,000.
To prepare for the building of the new stand, the buildings behind the current stand - the administration block, the shop and nursery - would be knocked down, with temporary facilities being provided during construction.
The temporary buildings will be occupied in October, with the demolition beginning the following month.
Construction of the new stand will start over and behind the current stand during next season before being completed during the 2017 close season.

Flynn's side are 22nd in League Two, two points above Hartlepool in the final relegation place.
A win will ensure the Exiles remain in the EFL, but if Pools beat title-chasing Doncaster Rovers, they could return to the National League.
"If we win, it's going to be the biggest party in Newport for a long time," Flynn told BBC Radio Wales.
He continued: "I'm not going to pull any punches or dress it up, it is the biggest game [in their modern history] for so many reasons. It means so much to the club and amazing fans and to me."
Newport were beaten 2-1 by Carlisle United on Saturday after leading in the 12th minute in a game which could have wrapped up their EFL survival.
Flynn says the game against Notts County will be difficult to watch but believes his side can take the win they need to stay up.
"It's going to be horrendous, it's going to be nerve-wracking but I really think the boys are going to deliver because they've been brilliant and I can't ask for much more so I'm quietly confident but it will be a stressful time," he said.
"We've had some unbelievably tough games considering the league and position we were in but they've answered everything and every time we've lost, we've had a good response so I'm hoping we'll get one on Saturday."
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Fans group the Foundation of Hearts has voted overwhelmingly in favour of helping to fund the Tynecastle club's new £11m main stand.
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Newport County boss Mike Flynn says their game against Notts County is the biggest in the club's recent history.