On a memorable evening for Swiss football, the so-called Nati repeatedly beat the odds, coming from 3-1 down with two goals in the final five minutes to force extra time and eventually penalties.
It was Switzerland's first knockout stage victory at a tournament since 1938 and the first time they have made the last eight since the 1954 World Cup which they hosted.
"Honestly I am still in shock," said Sommer, who became the most capped Swiss keeper with 65 international appearances.
"We showed courage, heart, we left everything out there. When you come back from two goals down against the world champions it is just unbelievable, and then to win on penalties, I could not be prouder of the way we did it."
France, who had started with an unusual three-man defence and lacked any real bite up front in the first half, looked to be heading for victory when Karim Benzema struck twice in two minutes early in the second half.
He latched on to Mbappe's through ball in the 57th minute to cancel out Switzerland's first-half lead from Haris Seferovic's header.
Benzema then headed home from near the goal-line two minutes later to take his tournament tally to four as France grabbed control of the game only minutes after the Swiss missed a 55th minute penalty and the chance to go 2-0 up when Hugo Lloris saved Ricardo Rodriguez's spot kick.
By the time Paul Pogba curled a sensational shot into the top corner in the 75th minute to make it 3-1 for France, who had reverted to a four-man backline in the second half, few believed the momentum could shift once more.
"Nobody believed in us anymore at that stage," Sommer said. "We felt France had become a bit complacent and maybe thought they had already won it. So we used that to our advantage."
Seferovic sneaked into the box again to head in his second goal with five minutes left and substitute Mario Gavranovic beat Lloris to snatch a 90th-minute equaliser and force extra time after France substitute Kingsley Coman's drive hit the bar.
Olivier Giroud twice came close for France having come off the bench but the game was to be resolved on penalties where Sommer swatted Mbappe's spot kick away after five Swiss players had scored to earn their first ever tournament shootout win.
Only 23 June 2021 has produced more goals (18 in 4 games) on a single day in European Championship history than the 14 goals scored in today's two games.
France have failed to reach the quarterfinal stage of a major tournament (Euros/World Cup) for the first time since the 2010 World Cup.
Switzerland have reached the quarterfinals at a major tournament (Euros/World Cup) for the first time since the 1954 World Cup, while this was their first-ever penalty shootout victory in either competition.
Switzerland have scored 3+ goals in consecutive major tournament matches (Euros/World Cups) for the first time since the 1954 World Cup.
There were just four minutes and three seconds between Hugo Lloris’ penalty save and Karim Benzema’s second goal of the game to put France 2-1 up.
Karim Benzema became only the second player to score 2+ goals in consecutive games with France at the European Championships after Michel Platini in 1984 (3 goals v Belgium and 3 goals v Yugoslavia in the group stage).
Hugo Lloris became the first goalkeeper to save a penalty for France in a major tournament game (Euros/World Cup) since Fabien Barthez in 2004 v England (exc. shootouts).
Four of Paul Pogba's last five goals for France have come from outside the box, including each of the last three.
Switzerland’s Steven Zuber provided his fourth assist at Euro 2020, no player has ever provided more at a single European Championships tournament (since 1980).
Haris Seferovic has scored in consecutive major tournament games (3 goals), after scoring just once in his previous 13 appearances across European Championships and World Cups.
N’Golo Kante made his 50th appearance for France, with the midfielder ending on the losing side in just four of his 50 games for the national team (W35 D11).
Spain eliminate Croatia in eight-goal thriller
Earlier in the day, Spain striker Alvaro Morata enjoyed sweet redemption by helping fire his side to a 5-3 win over Croatia after extra time on Monday to reach the quarter-finals of Euro 2020 in an enthralling match laced with drama and storylines.