France reached the last 16 and clashed with Switzerland at the National Arena in Bucharest, Romania, on Monday.
However, the 2018 World Cup champions were stunned in the first half after Haris Seferovic fired Switzerland into the lead.
Real Madrid striker Karim Benzema scored a brace in the second half as France took the lead against Switzerland.
Pogba, who has been in scintillating form for France, added his name to the scoresheet with a sensational long-range strike past Yann Sommer.
France's hopes of sealing a 3-1 win were dashed when Switzerland mounted a successful comeback through Seferovic and Mario Gavranovic.
Neither France nor Switzerland could break the 3-3 deadlock in extra time, forcing the round-of-16 match to head to a penalty shootout.
Pogba, Olivier Giroud, Marcus Thuram and Presnel Kimpembe all stepped up and converted from the spot-kick.
However, Kylian Mbappe's penalty was saved by Sommer as Didier Deschamps' men crashed out of the Euros.
Pogba has now taken to Instagram to congratulate Switzerland on their win, despite the "sadness" that the result brought to the France players.
The 28-year-old Manchester United star vowed that the France players would keep their "heads up" and "come back stronger."
Pogba wrote to his 45.4m followers: "Sometimes football can be cruel... cruel and also beautiful.
"The game brought us sadness but happiness to our opponents. That's the beauty of football. Of course, we all wished that we could have had a positive result.
"Thank you so much to all of our fans across the globe. It was beautiful to see you, to hear you, and to celebrate with you.
"You gave us hope and joy throughout all of our games. We will keep our heads up and we will come back stronger.
"Finally, I want to wish congratulations to Switzerland."
Premier League legends Roy Keane and Patrick Vieira were both critical of Pogba after France's devastating Euro 2020 exit.
Mbappe has broken his silence by posting a social media message to France fans, as he vowed the Blues would come back "even stronger for the next challenges."
Meanwhile, former Tottenham manager Jose Mourinho has identified a major mistake Deschamps made in France's penalty shootout defeat to Switzerland.
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