Kylian Mbappe's first-half strike put PSG in the driving seat with a 2-0 lead in the aggregate scoring but Benzema scored three goals after the break as Madrid claimed a 3-2 victory.
"We wanted to see this stadium like the fans were today, we needed them, the win is for them; it was very difficult, we pushed until the end and we deserved to go through," Benzema said postmatch.
"We lost the first leg, we were 1-0 down at half-time, it was a huge effort, the fans pushed us to give everything until the end. They're Champions League games, it's difficult. We know PSG like to have the ball. We started well, we had chances, we didn't take them and they scored on the counter, but in the second half we played our game.
"When we press we can beat any team, it's about pressing as a team, everyone in their position, and with the fans. Every game is a final for us, in the league and the Champions League. Today Real Madrid are alive."
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Both teams started brightly but it was PSG who enjoyed the early chances and Madrid goalkeeper Thibaut Courtois had to deny Mbappe on two occasions.
After 25 minutes, Gianluigi Donnarumma produced a stunning fingertip save from Benzema's long-range effort.
Less than 10 minutes later, Mbappe thought he had given the visitors the lead on the night but his goal was ruled out with Nuno Mendes in an offside position.
Madrid were unable to prevent Mbappe from opening the scoring five minutes later when he beat the offside trap and slotted the ball past Courtois.
Shortly after the break, Mbappe thought he had scored again but his stunning finish was disallowed for offside once more.
Just after the hour mark, Benzema capitalised on Donnarumma's poor error to make it 1-1 on the night.
Vinicius Junior had a great chances to level the scores on aggregate 10 minutes later but blazed his close-ranged shot over the bar.
Benzema made no mistake minutes later to make it 2-2 aggregate and seconds afterward, he completed a sensational hat trick when Madrid punished PSG's sloppiness from kickoff.
The France international scored his 309th goal for Madrid in all competitions, passing Alfredo Di Stefano for third most in club's history.
Despite reaching the semifinals last season and the 2020 final, PSG are still in search of winning the Champions League for the first time in its history.
"It's a disgrace that in 2022 a goal like that stands, it was a clear foul by Benzema on the goalkeeper," Mauricio Pochettino told a news conference. "After an injustice like that the team had a mental breakdown. It's a shame."
Speaking to Movistar after the game, Madrid boss Carlo Ancelotti said: "It's special above all because we won after a difficult game. It got more complicated after Mbappe's goal. We suffered a lot to get the ball back. We pressed them, we'd practised that in training, and then we got the first goal and the game changed totally, the fans started to believe more and more, it was spectacular, we're obviously very happy.
"We started well, with energy, then little by little we had more difficulties to press, they use the ball very well, they have players with quality, we suffered but that was normal. We took some risks, sometimes we left Mbappe one-on-one and that can make your life complicated. We needed the goal. After that the atmosphere in the stadium changed completely. The last half hour was spectacular."