Luciano Monzon's first-half penalty proved the difference between the teams as Nice edged towards Ligue 1 survival and deepened nine-man Auxerre's relegation fears.
The Argentinian defender converted from the spot in the 36th minute to earn the hosts three vital points, with Auxerre's cause not helped by the dismissal of Dennis Oliech for dissent three minutes later.
Willy Boly added to their woes when he too was sent off 12 minutes from time as Nice climbed four points clear of the drop zone and left basement boys AJA five points adrift of safety with five games remaining.
Auxerre had the better of the early exchanges and were within a whisker of taking an eighth-minute lead when Anthony Le Tallec's left-footed volley from Oliech's pass flashed just wide of the post.
Nice defender Francois Clerc had the woodwork to thank after misdirecting his headed clearance from Omar Kossoko's cross, the right post coming to his rescue in the 15th minute.
Roy Contout then saw a low volley well blocked by Nice goalkeeper David Ospina.
But, for all their early pressure, Auxerre were unable to find the breakthrough and Nice rubbed salt into their wounds nine minutes before the break.
The referee pointed to the spot following a foul by substitute Pyrecque Rizieyran on Franck Dja Djedje, and Monzon coolly sent the keeper the wrong way.
Oliech continued his protests too vociferously for the match official's liking and was sent off soon after.
Nice made their extra man pay after the interval. There were no further goals, but the hosts bossed possession.
They could have gone two up in the 64th minute when substitute Esmael Goncalves engineered a chance for himself down the right but Auxerre goalkeeper Olivier Sorin made a fine save, and again when midfielder Kafoumba Coulibaly fired just wide midway through the second half.
Boly's red card towards the end - for blocking Goncalves - left the visitors with an even bigger mountain to climb.
Elliot Grandin curled a superb shot inches over the bar in the 81st minute after getting the better of Rizieyran.
After dominating the second half, though, Nice almost threw away all their hard work six minutes from time. A mistake from Monzon saw the ball fall for Auxerre substitute Olivier Kapo, and his right-footed half-volley skimmed the bar on its way over.