Harry Kane tested Mosquera with a powerful strike from the inside-right channel before Tuchel prepared to alter his lineup.
During the transition, Quansah tangled with Rodríguez, received a yellow card, and had to limp off due to injury.
England opened the scoring while temporarily down to 10 players as Spence waited to substitute on.
Saka delivered a corner kick, and Jude Bellingham turned the ball in with his left foot from the edge of the six-yard box.
Panama nearly equalized immediately when Michael Murillo found space against a retreating English defense, but he hit his shot wide.
England capitalized on the error quickly when Bellingham found space down the left wing, waited for support, and crossed for Kane to head home from close range.
Despite looking vulnerable defensively, England held on as Tuchel introduced Eberechi Eze, Ollie Watkins, and Jordan Henderson.
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Panama saw a late goal by José Fajardo disallowed for a tight offside call, leaving Tuchel visibly displeased on the sideline.
