PSG were on the verge of another disappointing Champions League exit as they trailed Atalanta 1-0 into the 90th minute in Lisbon, where all the latter stages are being played after the long coronavirus shutdown.
But Mario Pasalic's 27th-minute strike for the Italians was finally cancelled out by Marquinhos, who scrambled in a last-gasp leveller.
Then, three minutes into injury time, Kylian Mbappe set up fellow substitute Eric Choupo-Moting to convert the clinching goal.
"The goals came very late but we deserved the victory. We never stopped believing or attacking," said Tuchel, whose team will face Atletico Madrid or RB Leipzig in the semi-final, the club's first since 1995.
"The boys who came off the bench had a big impact. We deserved it, we made our own luck."