In a frenetic encounter at Allianz Riviera, Alves' first Ligue 1 goal proved the difference in the 82nd minute, with Angel Di Maria having cancelled out Allan Saint-Maximin's 17th-minute opener.
Having stepped into the void left by the injured Neymar, Di Maria has been excellent for Unai Emery's side in recent weeks, and netted his 11th goal of the campaign in spectacular fashion 21 minutes in.
Hunting a fourth consecutive away league win, PSG stepped up the pressure after the interval, with Kylian Mbappe seeing a goal correctly disallowed for offside.
Nice still presented a threat on the counter, though - Balotelli heading against the crossbar with 15 minutes remaining.
But PSG would not be denied a ninth successive league victory as Alves had the final say, the Parisians near certainties to regain the title with a 17-point lead over Monaco at the top.
Buoyed by their 5-2 hammering of Guingamp last time out, Nice started on the front foot - Pierre Lees-Melou forcing Alphonse Areola into action, while Balotelli rattled the upright with a powerful header.
PSG responded well - Cavani denied by Walter Benitez before turning the rebound against the post from point-blank range, and that miss would soon prove costly.
Having netted four times against Guingamp, Alassane Plea turned provider with an exquisite throughball into the path of Saint-Maximin, who duly slotted past the onrushing Areola.
Nice's lead would last just four minutes though, as - having already seen one effort well saved by Benitez - Di Maria raced through the hosts' defence before prodding home.
Di Maria could have had an assist to his name when he whipped in a brilliant corner in the 34th minute, only for Adrien Rabiot to head wide from six yards out.
Layvin Kurzawa replaced Yuri Berchiche for the second-half restart, and Saint-Maximin wasted no time in getting the better of PSG's substitute, Areola rushing out to deny Nice's winger a second.
Mbappe had the ball in the back of the net just prior to the hour, but had just strayed offside before connecting with Di Maria's delicate pass.
That was followed by chances at either end - Benitez doing well to tip Alves' acrobatic attempt over while Areola kept out a rasping Balotelli strike.
Balotelli looked set to double Nice's tally when he connected with an in-swinging corner, but his header clipped off the bar, before Mbappe missed a golden opportunity at the other end.
His blushes were spared, however, as Alves leapt high above Nice's defence to flick Rabiot's pinpoint cross into the bottom-left corner.