Les Bleus looked on course to begin their campaign with a draw after Bogdan Stancu had cancelled out Olivier Giroud's opener, but Payet stole the headlines late on to get his side off to a winning start.
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Romania's qualifying matches en route to the tournament yielded just 13 goals, the lowest of the teams to have made it to France, but they almost took the lead after just four minutes when Stancu squandered a glorious chance to dampen the party atmosphere inside the stadium.
The opening came from a corner as the ball was flicked onto the back post, only for Stancu to fire straight at Hugo Lloris from point-blank range when it looked easier to score.
The hosts were undone by another corner moments later, however, as Romania hinted at a potential Achilles' heel for France when Florian Andone got in front of Giroud only to send his header onto the roof of the net.
The host nation have only won five of 17 previous opening games at European Championships, but France soon settled into the match and attempted to improve that statistic, with Giroud nodding their first clear chance past the post having been found by man of the match Payet.
They were even closer to breaking the deadlock just three minutes later when Bacary Sagna's cross struck Vlad Chiriches and deflected into the path of Antoine Griezmann, who reacted in time to steer a header against the post from close range.
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France's midfield trio began to have a growing influence on the game, and Paul Pogba almost provided the opening goal shortly before the half-hour mark when his long-range effort swerved over the crossbar.
Didier Deschamps had seen his side score 13 goals in their four games running out to the tournament, but they struggled to find the end product against Romania and Cristian Sapunaru kept things goalless with a timely challenge on Griezmann just as the forward looked set to convert Payet's low cross, sending the ball just past the post.
The first half was a story of missed opportunities, though, and Giroud, who had scored seven goals in his last five starts for France heading into Euro 2016, missed one more in stoppage time when he nodded Payet's corner over the crossbar.
France were almost made to pay for their wastefulness shortly after the restart as Romania made another quick start to the half, and once more it was Stancu who squandered the chance when he put the ball wide having beaten the offside trap to latch on to Nicolae Stanciu's chipped through-ball.
The hosts soon settled back into the game, though, and Payet began to weave his creative magic as the half wore on, setting up Giroud to have France's first shot on target of the evening when he fired a deflected effort straight at the keeper.
Payet was the creator again shortly afterwards when he danced into space to cross the ball before picking out Pogba on the edge of the box, but the Juventus man's first-time volley was too central and once again Ciprian Tatarusanu made the save.
The opening goal of the tournament did finally arrive just before the hour mark, though, and again the chance came from the boot of Payet as he floated a ball into the box that Giroud met, beating the keeper in the air to nod it into the unguarded net.
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The hosts were ahead for just seven minutes, though, as Romania quickly levelled things after being awarded a penalty for a clumsy foul on Stanciu from Patrice Evra.
Stancu, who had already missed two clear chances for the visitors, stepped up to take the spot kick and this time made no mistake, sending Lloris the wrong way and tucking the ball into the bottom corner.
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Anthony Martial was brought on in an effort to put France back in front, and he had a chance to do just that when he was teed up by Giroud, but he didn't catch his shot right and it was a comfortable save for the keeper in the end.
It looked as though France might become the latest host nation to slip up in the opening game of a European Championship, but Payet made himself the hero late on with a stunning strike into the top corner to win the match for his side.
The West Ham United playmaker collected the ball just outside the area before cutting inside and arrowing an unstoppable shot past a helpless Tatarusanu to send the Stade de France into raptures.
There was no time for Romania to respond as they fell to defeat for just the second time in their last 20 outings, while it is now five wins on the bounce for tournament favourites France.