The kickoff was brought forward to 1200GMT to ensure maximum exposure in Asia and PSG, who now have 83 points from 31 games, were keen to impress.
They failed to do so in a lacklustre performance but Alves's 82nd-minute header was enough for them to edge closer to a seventh French league title.
Nice, who hit the woodwork through Mario Balotelli in the second half, opened the scoring through Allan Saint-Maximin, whose early goal was quickly cancelled out by Angel Di Maria.
Later on Sunday, Olympique de Marseille lost ground in the race for second when Memphis Depay's last-minute goal earned Olympique Lyonnais a 3-2 win at the Stade Velodrome.
Third-placed Marseille, on 59 points and seven behind Monaco, opened the scoring courtesy of a Rolando first-half strike. However, Adil Rami's own goal and Houssem Aouar put Lyon in front.
Kostas Mitroglou equalised in the closing stages, only for Depay to strike again for the visitors, who are fourth on 57 points.
At the Allianz Riviera, Saint-Maximin beat PSG keeper Alphonse Areola from close range after being perfectly set up by Alassane Plea on 17 minutes.
That spurred PSG into action and the visitors equalised just four minutes later thanks to Di Maria, who found the back of the net with a left-footed effort.
Balotelli went close to restoring Nice's lead eight minutes before the break, but his curled effort shaved the far post.
PSG had a chance on the hour, but Nice keeper Walter Benitez pulled off a superb save to tip Alves's bicycle kick over the bar.
Balotelli forced Areola into another brilliant save two minutes later as Nice piled on the pressure.
The Italian hit the woodwork 15 minutes from time with a header, while at the other end PSG forward Kylian Mbappe headed over the bar.
Alves proved to be the most accurate with his head, meeting Adrien Rabiot's cross in the 82nd minute to beat Benitez and seal the win.
(Reporting by Julien Pretot; Editing by Christian Radnedge)
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