The U.S. missed a great chance to take the lead at 25 minutes when, after dispossessing Mexico in its own half, Sebastian Lletget sprang Paul Arriola in the area, but his shot came back off the post with only the keeper to beat.
Despite having the better of the shots and possession, Mexico failed to score as a scrappy U.S. side went into halftime tied 0-0.
Orbelin Pineda wasted two opportunities to put Mexico in front to start the second half: one from a wide-open shot in the penalty area and the second from close range on a ball in from the right flank.
The U.S. was undeterred though, and Matthew Hoppe had two good chances blocked away by the Mexican defense, while Arriola had a shot on the doorstep impossibly stopped by El Tri keeper Alfredo Talavera as the second half progressed.
A thrilling end to regular time saw both sides fashion decent chances without scoring and the game headed to extra time.
The extra 30 minutes continued the theme of the game: Plenty of action but no goals, until Robinson hooked on to a Kellyn Acosta free kick in the 117th minute and beat Talavera to give the U.S. a massive win.
The U.S., which secured its seventh Gold Cup title, beat Mexico in the inaugural CONCACAF Nations League final with a Christian Pulisic extra-time winner on June 6 in the teams' last meeting.
"When I look back at it, I'm most proud for the team, for the guys," U.S. head coach Gregg Berhalter said. "We were together for a month, and we just saw them growing, we saw the team growing, saw individuals growing and improving. We saw the team chemistry growing.
"All losses are painful, especially in circumstances like these," Mexico manager Gerardo Martino said after the match. "I believe we've competed well against the United States. We had really great moments in our favor tonight. This loss shouldn't be a catastrophe for the team, of course it hurts, but we'll look to improve.
"It's true we lost two finals, and it's true we've lost those games on set pieces, but I don't see why we should change our style of play. We didn't prepare this team for 2.5 years to play one way [in the Nations League and Gold Cup] and then to play a different way in the qualifiers."
The U.S. went into the match Sunday as the underdog because many of its top players from the Nations League tourney were not on the Gold Cup roster so they could prepare for their European seasons.
In fact, there were only three holdovers from the Nations League tournament compared to Mexico fielding eight starters Sunday, who also played in the Nations League final.
"Before the final today, there was a feeling inside of me that I just wanted it so bad for them, that they can taste this, and then they can win a trophy together," Berhlater said. "So really proud of the group, really proud of their resiliency."
Mexico, meanwhile, will look back on another missed opportunity, having spurned the chance to claim a 12th Gold Cup title, as well as suffering just its second defeat in the tournament's final.
"We were aware that [the United States] could hurt us on set pieces and they did, again," Mexico veteran Hector Herrera said. "I don't agree [with the honeymoon with Martino being over]. We have to evaluate ourselves and reload for qualifying. If we win, we're criticized. If we lose, we're criticized.
"It definitely hurts [to lose]. But I think we're calm and we're good though because we dominated throughout -- we just couldn't score."