The host national had led through Olivier Giroud’s header just after the break, only for the visitors to equalise from the spot through Bogdan Stancu.
The game looked to be heading for a stalemate, until Payet’s moment of brilliance in the dying minutes.
Romania started well and should have had an early lead through Stancu, who found himself unmarked at the back-post in the fourth minute to connect with a flicked-on header, but his shot from point-blank range was straight at Hugo Lloris.
Florin Andone headed onto the roof of the net shortly afterwards, but slowly France made their way back into the game, with Antoine Griezmann’s 10th-minute shot blocked by Vlad Chiriches, before Giroud headed Payet’s pinpoint cross wide.
On 14 minutes Griezmann came closest to a goal; his header hitting the upright from close range.
The Atletico Madrid forward fired a low drive agonisingly wide 10 minutes before the break, before Giroud squandered another header, again from Payet’s delivery.
Romania started the second half like a house on fire, but again Stancu missed an early chance – volleying wide after beating the offside trap.
Giroud fired straight at the goalkeeper shortly afterwards, before Ciprian Tatarusanu pulled off a great save to keep out Pogba’s powerful volley.
The goal eventually came though, and it was Giroud who provided it – his glancing header from Payet’s cross finding the back of the net on 57 minutes.
However, the lead lasted all of eight minutes as Patrice Evra conceded a clumsy foul on Nicolae Stanciu, with Stancu converting the spot-kick.
Anthony Martial was brought on for Pogba and almost delivered straight away, but his powerful shot was straight at the grateful Tatarusanu.
However, there was one more chance – a half-chance at that – but that was more than enough for the brilliant Payet, who flashed a left-footed curler past the keeper from the edge of the area with a minute left on the clock to hand France a winning start.
France: Lloris, Sagna, Rami, Koscielny, Evra; Kante, Matuidi, Pogba; Griezmann, Giroud, Payet.
Romania: Tatarusanu, Sapunaru, Chiriches, Grigore, Rat; Hoban, Stanciu, Pintilii; Popa, Andone, Stancu.