France midfielder Menez set up Christophe Jallet's opener and netted from a Nene cross in the 76th minute after Dennis Oliech had equalised for the visitors.
Nene added a third with a penalty nine minutes from time and Dariusz Dudka scored a late goal in the closing stages as second-placed PSG narrowed the gap to three points on leaders Montpellier who crushed Lorient 4-0 on Saturday.
PSG now have 33 points from 16 games and Kombouare, who looked on the brink of losing his job after three successive defeats in all competitions, may not have to look over his shoulder, at least until next weekend.
Elsewhere, Girondins Bordeaux beat Nancy 2-0 to move up to 12th on 19 points and the visitors stayed second from bottom with 14 points.
Olympique Lyon, in fourth, kept their Ligue 1 campaign alive with a 3-2 home win over sixth-placed Toulouse.
Bakary Kone headed home before the break, Ederson found the back of the net with a superb curled shot and Lisandro Lopez converted a penalty in the second half to give Lyon 29 points.
Umut Bulut and Moussa Sissoko pulled two back in the 68th and 75th minutes but Lyon then held on.
PSG also struggled against Auxerre.
"I am proud of the players' behaviour, they showed character," PSG coach Kombouare told a news conference aired on French TV channel Infosport.
"We won today because they disregard what's being said about the club, about my situation."
Brazilian Nene told French TV channel Foot Plus: "It was a great game from the whole team. We showed character, which is what we lacked in our last games.
"It was tough but at least we won and we found the back of the net. We are all together in this and the coach has been doing his best. People talk but we have to stay away from this. The coach is doing a good job."
At the Parc des Princes, Kombouare sat on the bench despite speculation suggesting he was to be replaced before the match by former France midfielder Claude Makelele, now a sports adviser at the club.
After a lacklustre opening half, PSG piled on the pressure and were rewarded for their efforts in the 52nd minute.
Midfielder Jallet fired home from a Menez cross to make it 1-0 but Auxerre reacted quickly.
Oliech had a fierce shot from just inside the box parried away by Salvatore Sirigu but the Kenya striker had followed up and poked the ball home to equalise in the 59th minute.
Menez put PSG back in command 14 minutes from time when he latched on to a cross from the left by Nene, who converted a penalty in the 81st minute after being fouled in the box by Cedric Hengbart.
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