The action-packed second half proved to be a nightmare for Lyon, who saw Bafetimbi Gomis miss a chance to make it 2-2 from the penalty spot with three minutes to go then had centre back Dejan Lovren sent off in stoppage time for a foul in the box.
Mutu, who had already put his side ahead with a close-range header in the 65th minute, netted the resulting penalty to seal the home team's victory.
Gaaetan Courtet scored the opener for relegation-threatened Reims just after the half-hour mark to spark hopes of a fifth win of the season, but second half goals from Souleymane Camara, Marco Estrada and Younes Belhanda gave Montpellier an easy win which keeps them in the hunt for European football next season.
Lyon are on 45 points from 23 games, with PSG on 48 after the capital club thrashed Toulouse 4-0 away on Friday. Third-placed Olympique Marseille can draw level on points with Lyon if they beat bottom club Nancy later on Sunday.
Lyon dominated possession for an hour but struggled to threaten goalkeeper Guillermo Ochoa until Alexandre Lacazette, who had a first-half goal disallowed for offside, opened the scoring with a superb volley in the 53rd minute.
Their lead did not last long, however, as Chahir Belghazouani converted Ajaccio's second effort on target four minutes later to make it 1-1.
The Morocco international, who had just returned from African Nations Cup duty, initiated 14th-placed Ajaccio's second goal when he ran down the left wing to set up Johan Cavalli, whose neat cross was headed home by Mutu.
Lyon increased the tempo and thought they would salvage a point in the 87th minute when Gomis was awarded a penalty after Felipe Saad fouled him just inside the box, but Ochoa saved the France striker's effort.
The action was not yet over as Lovren was shown a red card for a foul on Dennis Oliech in stoppage time and Mutu converted the ensuing penalty to score his seventh league goal of the season with Ajaccio recording a first win since mid-December.