PSG looked on course of the last eight when Kylian Mbappe put them 2-0 up on aggregate late in the first half.
Los Blancos roared back in the second period, though, thanks to Benzema's hat-trick – the second and third of which came within the space of just 106 seconds.
It saw him become the oldest scorer of a hat-trick in Champions League history (aged 34 years and 80 days), 17 days older than Olivier Giroud in December 2020 for Chelsea against Sevilla.
The treble also saw him move above Alfredo Di Stefano into third on Madrid's all-time top goalscorer list with 309.
Ancelotti was full of praise for the talismanic striker as Madrid overturned a first-leg deficit in a Champions League knockout tie for only the fourth time.
"Karim Benzema is a fantastic leader, a fantastic centre forward," he said.
"I'm very happy with what he's doing and his attitude.
"After the first goal, we got into a good dynamic, we pressed and controlled the ball better. We had fewer difficulties in the second half compared to the first.
"At half-time, we said to ourselves that we had to stay in the game, even if it was difficult, thinking that something could happen.
"PSG are a great team with great players, but that's football, every little thing can change everything like that first goal."
PSG boss Mauricio Pochettino was adamant Madrid's first goal should have been ruled out for a foul by Benzema on goalkeeper Gianluigi Donnarumma, but Ancelotti did not agree with his opposite number.
Asked if it was a foul, the Italian said: "I don't think so.
"We suffered a lot, but we held on. It was difficult to hold the ball, we tried to press then Karim's good pressure gave us the opportunity to win the game. During the last half hour, there was only one team on the field."
Benzema's third sparked jubilant scenes at the Santiago Bernabeu, with Ancelotti describing the stadium as "magic".
"I have played many games in this competition and many difficult ones like this," he added. "This stadium has magic; it has a very beautiful history and that is what happens. Hopefully it will not be the last night like this."
Madrid returns to domestic action on Monday when it visits Real Mallorca.