After four successive defeats - three in Europe - since replacing Derek McInnes in June. the Belgian boss oversaw a four-goal win at Tynecastle.
But Hearts' embattled supporters were treated to an eye-catching glimpse of the type of front-foot football former Sint-Truiden boss Vrancken intends to implement as his new-look team tore United apart in their first home game of the Scottish Premiership campaign.
Tom Renaud, Calvin Miller and Josh McPake all marked their first league outings at Tynecastle with goals, while last season's talisman Claudio Braga broke his duck for the campaign.
Vrancken, without several key defenders, named an ultra-attacking line-up with no defensive-minded midfielders, midfielder Oisin McEntee at centre-back and winger Miller deployed at left-back.
Among five changes to the team that started Thursday's 6-1 Europa League drubbing away to Benfica, Greek winger Alexandros Kyziridis dropped out of the squad entirely as he prepares to seal a £1.7m transfer to a Turkish club.
United boss Jim Goodwin made one change to the side that kicked off the campaign with an impressive 1-1 draw at home to Rangers as Daniel Bennie replaced Zac Sapsford in attack.
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The Jambos made a fast start as Renaud, a few days after notching against Benfica, unleashed a ferocious strike beyond Jack Walton from just short of 30 yards after being teed by Braga in the fourth minute.
Settled by their early opener, Hearts went in search of a second and Braga saw a stooping header, from Amadou Ba-Sy's cross, clawed away by Walton.
The hosts were gifted the chance to double their lead when Bennie wrestled Jamie McCart to the ground while trying to defend Miller's corner.
Following a VAR check, referee Kevin Clancy was called to the monitor and duly awarded a penalty. Miller stepped up to fire the penalty straight down the middle in the 12th minute.
United, stung by their chastening start, started to feel their way into the game and Lachlan Rose saw a close-range header deflected behind while Dylan Tait had a shot charged down in the box by McCart.
Hearts almost went further ahead in the 32nd minute when Ba-Sy's low strike was pushed out by Walton before Braga shot against the post, with Renaud firing wide from the rebound.
At the other end, Rose's header was cleared over the crossbar by Renaud and then Bennie blazed over after United worked their way into the Hearts box.
But Hearts killed off their hopes of a fightback in the 56th minute when Braga drilled in a low angled shot from 12 yards out after being set up by Sabah Kerjota.
McPake then marked his first competitive start for the Jambos with his first goal as he tucked a clinical low finish beyond Walton after playing a one-two with Braga on the left of the box.
