Suarez was the only player to miss in the shoot-out as Peru advanced 5-4 on spot kicks after a 0-0 draw in Salvador on Saturday.
The Barcelona forward was in tears after the defeat as Uruguay – unbeaten in the group stage – bowed out.
Suarez's former club team-mate Neymar, who is linked with a return to Barcelona from Paris Saint-Germain, offered the Uruguayan support.
"Get up," the injured Brazil forward wrote on his Instagram story, alongside a video of Suarez crying.
"You are very great, brother. I love you friend."
Neymar is widely reported to be returning to Barcelona ahead of next season, two years after leaving the La Liga champions.
And while Suarez's missed penalty will be the main talking point, it was not the only failing of the day for Uruguay, who were their own worst enemies on a night when Peru became the first side in the Copa America not to register a shot on target.
The Uruguayans were caught off-side seven times against Peru, the most in a game of the 2019 Copa America, with two of those calls ruling out goals.
Suarez saw a strike ruled out for offside himself, though the margins were incredibly slim.
The match also continued a trend of sloppy play in Uruguay matches.
Uruguay have now been involved in the joint-most shootouts in Copa America history with nine. And the loss against Peru saw the side slide into a losing record in those matches, with Saturday's contest marking the fifth loss in those nine matches.
Suarez and Uruguay will only have to wait one year for a chance at redemption, with Copa America switching to even-year tournaments in line with the Euros in 2020.