The 29-year-old entered the fray as a 96th-minute replacement and went on to play a crucial role in his country’s stunning win, tucking away his penalty after 120 minutes of breathless action.
Haris Seferovic’s perfectly-placed header had given Vladimir Petkovic’s side a surprise lead quarter of an hour into the tie in Romania, with Magpies man Schär named among the substitutes.
Ten minutes into the second half, Switzerland squandered a huge opportunity to extend their lead, but Ricardo Rodriguez’s tame penalty was kept out by Hugo Lloris.
The Spurs keeper’s save proved to be a major turning point in the contest, as barely a minute later Karim Benzema’s neatly-taken equaliser levelled matters.
And then just another minute on, the Real Madrid forward’s header from close range turned the game upside down with 30 minutes still left to play.
With quarter of an hour left, Paul Pogba’s sensational strike from 25 yards made it 3-1 to Didier Deschamps’ men, and the game looked to be effectively over.
But Seferovic nodded home to set up a tense finale, and with their hopes slipping away, Mario Gavranovic’s fine finish took the game into extra time in dramatic fashion.
Schär was summoned from the bench six minutes into the first period of extra time, replacing the influential Seferovic, but neither side could find a winner in the 30 additional minutes.
The Newcastle defender took his side’s second penalty in the ensuing shoot-out, calmly sidefooting home.
Both sides netted their first four penalties before Admir Mehmedi tucked home Switzerland’s fifth, meaning France had to score.
But Kylian Mbappe’s spot-kick was sensationally kept out by Yann Sommer, sealing one of the shocks of the tournament so far and sending Schär and his team through to the last eight, where they will play Spain.
Meanwhile, Emil Krafth could be in action on Tuesday evening.
The full back, United’s only other representative left in the tournament, and his Sweden side take on Ukraine in their last 16 clash on (kick-off 8pm BST).