Gameiro scored twice in the space of four minutes after the break after Carl Medjani had cancelled out the France striker's second-minute opener.
PSG have 23 points from 10 games after notching their fourth consecutive league win, with Montpellier and Olympique Lyon on 20 in second and third place respectively.
Stade Rennes are fifth on 18 points, one behind champions Lille, after goals by Jires Kembo-Ekoko and Jonathan Pirtoipa in each half gave them a 2-0 victory at home against Lorient in a Brittany derby.
Ajaccio remain second from bottom on seven points.
"It's always nice to score a hat-trick but the most important thing tonight is that we won," Gameiro told French TV channel Foot Plus.
"These goals are a bit like [Filippo] Inzaghi's, I'm not used to scoring goals like this," he added, referring to the Italian striker's close-range efficiency.
"We had a hard time in the first half but we came out on top. Let's continue like this and maybe we can hope to win the title."
PSG, who have not won the French league since 1994, made a flying start with Gameiro opening the scoring from close range after Sylvain Armand had deflected a Nene corner.
Ajaccio hit back in the 24th minute when Medjani fired home from inside the area and came close to a second on the stroke of half-time but Salvatore Sirigu pulled off a spectacular save to deny Richard Socrier.
PSG, however, had much more class and took control soon after the restart.
In the 50th minute, Mohamed Sissokho headed a Christophe Jallet cross into the path of Gameiro who poked the ball home.
Three minutes later, Gameiro latched onto a back-pass from Clement Chantome to make it three.
Elsewhere, 19-year-old Cameroonese striker Vincent Aboubakar netted twice to help Valenciennes beat Sochaux 3-0.
