PSG had talisman Zlatan Ibrahimovic sent off in the first half and fell behind to a Gary Cahill goal with nine minutes remaining, only for Chelsea old boy David Luiz to take the tie to extra time with an emphatic 86th-minute header.
Silva appeared to have gifted Chelsea a place in the last eight after conceding a penalty for handball that Eden Hazard converted, but the PSG captain scored with six minutes of the game remaining to send Laurent Blanc's side through on away goals after a 3-3 aggregate draw.
"If you analyse both matches, being fair, PSG's qualification is deserved," Blanc said.
"We tried to play more football than Chelsea and we created more dangerous chances. It was a fantastic performance by the whole squad."
The irony of the scoreline will not have been lost on PSG, who were eliminated by Chelsea on away goals in last season's quarter-finals after another 3-3 draw, but their breakthrough victory was richly deserved after a mature and resolute display on a tense and fractious night.
The French champions' dreams of an unprecedented quadruple remain intact, while Jose Mourinho will have to wait at least another season for an opportunity to win the Champions League with his beloved Chelsea.
"Our performance was not good enough," the Chelsea manager told Sky Sports.
"The opponent was stronger than us. They coped better with the pressure of the games. Because they had 10 men, we felt more the pressure of winning and they had nothing to lose."
It was a game almost totally devoid of incident until a moment of controversy in the 32nd minute that thrust referee Bjorn Kuipers beneath the spotlight beams.
After a PSG move broke down, Ibrahimovic showed a shade too much eagerness to regain possession by sliding in on Oscar, but although he caught the Brazilian on the ankle, both players were equally committed to the challenge and the red card that followed felt excessive.
PSG were, understandably, rattled, with Thiago Motta booked for dissent, and Chelsea threatened to capitalise, with Diego Costa denied a penalty after clearly being caught on the lower leg by Edinson Cavani.
But PSG kept their shape well and shortly before the hour they came within inches of going ahead.
– Verratti incurs suspension –
A brisk move down the centre of the pitch involvingMarco Verratti and Javier Pastore saw the latter release Cavani, who neatly skirted Thibaut Courtois, only to hit the left-hand post from an angle with the goal at his mercy.
PSG continued to press, Cahill blocking from Blaise Matuidi and Courtois saving from Pastore, and with the tie on a knife-edge, tempers began to fray.
Costa was booked for a dangerous late challenge on Silva and in the aftermath, Luiz received a yellow card for trying to claim he had been headbutted by the Chelsea striker.
Moments later Verratti, too, saw yellow, for fouling Hazard, ruling him out of the first leg of the quarter-final.
After Sirigu had thwarted Ramires at his near post, Chelsea broke the deadlock from a corner when Costa took an air shot and the ball fell kindly for Cahill to slam home.
But after PSG substitute Ezequiel Lavezzi had seen a header saved by Sirigu, his corner allowed Luiz to take the tie to extra time with a thumping near-post header that cannoned into the net via the crossbar.
Silva's recent performances have suggested a return to his optimum level, but he was guilty of a rush of blood to the head early in extra time when he challenged Kurt Zouma for a high ball with his arms way above his head.
Replays were inconclusive as to whether he actually made contact with the ball, but Hazard made no mistake from the spot.
Courtois saved brilliantly from a Luiz free-kick and a Silva header, but the PSG captain had the final say with a goal that settled a breathless and dizzying contest and took the visitors into a new dimension.
"The goal at the end was incredible," Silva told beIN Sports. "For me, for this team, for the club."