Sir Alex Ferguson's side suffered an early fright in the region best known for providing the supposedly vampire-filled setting for Bram Stoker's Dracula.
But Pantelis Kapetanos's early goal for Cluj didn't suck the life out of United, who maintained their 100 percent start to this season's competition thanks to van Persie's brace.
United were rarely at their best at the Dr Constantin Radulescu Stadium, but Cluj ultimately lacked the quality to take advantage and Ferguson could take heart from the way van Persie and Wayne Rooney combined for both goals.
"The pass and the finish were equally good for Robin's second goal. It was a fantastic finish," Ferguson said.
"But they kept us under pressure and on edge because we didn't finish them off.
"To be fair we defended well apart from the goal. I've got to be pleased because Cluj have a good home record and coming away from home in Europe is never easy."
Ferguson's star strikers were starting together for the first time, with Javier Hernandez also included in an attacking line-up, as United looked to bounce back after Saturday's surprise 3-2 home defeat against Tottenham.
United's go for broke approach worked, but only after Cluj made a flying start.
The Romanian champions had won their opening game in Braga a fortnight ago thanks to two goals from Rafael Bastos and they showed no signs of being overawed by United in the early stages.
Ioan Andone's side gave a warning of their ability as Modou Sougou headed just wide from a Bastos corner.
United's defensive frailties have been clear for all to see this season, but it was still shocking to witness the ease with which Cluj sliced through the visitors' rearguard to take the lead in the 14th minute.
When Bastos flighted a long pass over the head of Patrice Evra, the United left-back's lack of pace was brutally exposed as Sougou surged clear before drilling over a low cross.
Even then the danger could still have been averted but Jonny Evans had taken up a position too far from Kapetanos and the Greece forward took full advantage to sweep his shot past David de Gea.
Rooney led United's response and the England striker's snap-shot forced Cluj goalkeeper Mario Felgueras to concede a corner with a sprawling save.
Anderson flashed a powerful effort narrowly wide moments later and, with Cluj fading as an attacking threat after Sougou went off injured, United's pressure paid off with a 29th minute equaliser.
While United could claim their dominance at that point deserved a reward, the actual goal had more than a hint of good fortune about it as van Persie mistimed his attempt to head goalwards from Rooney's free-kick, but still found the net as the ball looped over Felgueras off the Dutch forward's shoulder.
Van Persie had a chance to strike again before the break but his shot was well blocked by Felgueras.
The former Arsenal star didn't have to wait long to notch his second of the night as United took the lead four minutes into the second half.
Van Persie's first had been a little fortuitous, but the second bore all the hallmarks of the Dutchman's sublime talent.
Rooney was the creator once more as his lofted pass picked out van Persie's run and the United forward, noticing that Felgueras had advanced too far off his line, cleverly flicked the ball past the goalkeeper and into the unguarded net with the outside of his boot.
De Gea preserved United's advantage as he tipped over Cadu's powerful header from a Bastos cross.
Some Cluj fans took to directing green laser pens at United's players in a bid to distract them in the closing stages, but De Gea remained focused and produced an excellent stop to deny Bastos in stoppage-time.