Juventus winger Weah was sent off in the 18th minute after an off-the-ball clash with Panama's Roderick Miller in a stormy Group C battle at Atlanta's Mercedes-Benz Stadium.
Although Folarin Balogun fired the US into the lead four minutes after Weah's dismissal, Panama's extra man advantage ultimately took its toll and the Central Americans hit back with goals from Cesar Blackman and Jose Fajardo to seal victory.
The defeat means the United States will almost certainly have to win or draw against group leaders Uruguay in their final group game next Monday to have any chance of advancing to the knockout rounds.
Uruguay all but assured themselves of a place in the last eight with a thumping 5-0 win over Bolivia in East Rutherford on Thursday.
In Atlanta, meanwhile, US coach Gregg Berhalter said after his team's defeat that Weah's red card had been the decisive moment of the match, describing the sending off as "silly."
"The match-changing event is obviously the red card and it puts us in a tough spot but we expected that from them," Berhalter said.
"I can't fault the effort of the group, especially after going down a man. The guys dug in and we were close to coming out with a point. But it's a shame, because there was more in this game, and a silly decision by Timmy leaves us shorthanded."
Weah, meanwhile, took to social media to apologize for his red card.
"A moment of frustration led to an irreversible consequence, and for that, I am deeply sorry to my teammates, coaches, family and our fans," he wrote on Instagram.
Weah 'sad and angry'
"I sincerely apologize to everyone. My love for this team goes beyond just football and I'm so sad and angry at myself for putting my brothers through what they went through tonight."
Berhalter's team might have snatched the lead in the 81st minute after Weston McKennie's fine run and cross found substitute Ricardo Pepi at the far post. Pepi's tame header fell into the arms of Panama goalkeeper Orlando Mosquera, however, to leave it at 1-1.
That miss proved costly, as with seven minutes remaining, Fajardo swept in an emphatic finish from Abdiel Ayarza's low cross to put Panama 2-1 ahead.
A fractious finale saw Adalberto Carrasquilla sent off after a cynical hack on US captain Christian Pulisic in the dying minutes as Panama hung on for the win.
"That's soccer at the end of the day. We know what we signed up for and playing in a Copa America, we knew it was going to be a fight," US midfielder Tyler Adams said afterwards.
"Credit to Panama, they did their job and they got the result. Obviously, I think moving forward, we need to control our emotions in certain situations.
"The team fought for everything after we got the red card. Definitely can't fault the effort. But by winning the first game we've put ourselves still in a position that we have all to fight for in the next game."
Adams said Weah had apologized for his rash dismissal following the game.
"You never mean to get a red card, under any circumstances," Adams said of Weah. "He's not that type of person.
"He apologized to the team. And you know, just respect to the rest of the team because they fought for every single ball, every single duel, every single minute, and we still created chances even after going down to 10 men."
Uruguay, meanwhile, looked in ominous form in Thursday's other game, demolishing Bolivia with five unanswered goals at the MetLife Stadium.
Facundo Pellistri, Darwin Nunez, Maximiliano Araujo, Federico Valverde and Rodrigo Bentancur were all on target for the Uruguayans.
