Vladimir Petkovic's men were 1-0 down in just over four and a half minutes as Sebastian Lletget capitalized on Switzerland's failure to clear their lines.
His close-range strike made it five goals in his last six USA games for Lletget, but the visitors' joy was short-lived as Ricardo Rodriguez leveled in the 10th minute.
SEBASTIAN LLETGET.— ESPN FC (@ESPNFC) May 30, 2021
GOAL FOR THE USMNT ‼️ pic.twitter.com/NVsHkqFmBb
Rodriguez found the net with a deflected drive from the edge of the area following Silvan Widmer's right-wing cross, but the left-back then followed his goal by spurning a gilt-edged chance to complete the turnaround.
The Torino left-back sent a penalty wide of the right-hand post after Sergino Dest was ruled to have handled in the area in the 41st minute.
Rodriguez's blushes were spared after another defensive error from Barcelona player Dest presented substitute Steven Zuber with a simple 63rd-minute chance that did not go begging.
Switzerland, who have won five straight matches, host Liechtenstein on Thursday in their final warm-up game before beginning their Group A campaign against Wales in Baku on June 12.