The Rio silver-medallists looked dead and buried after Danny Barrett crossed over for the first try and Perry Baker added a quickfire double, with captain Tom Mitchell forced off with an injury.
Ollie Lindsay-Hague clawed back the deficit just before the break, spotting a gap and darting over the whitewash for a half-time scoreline of 21-7 to the US.
Ben Harris gave GB further hope early in the second period after the US lost a man to the sin-bin, the 21-year-old taking full advantage to score a second try.
With a man advantage and fewer than five minutes remaining, Alex Davis went under the posts to level the scores and set up a nailbiting finish to the quarter-final clash.
Dan Norton, the leading all-time try scorer in rugby sevens history, added to his impressive haul following a quick turnover penalty and although the conversion was missed, Britain held on for an astonishing comeback win and a mouthwatering semi-final clash against New Zealand.
Speaking to the BBC after the match, a delighted Norton said: "I'm struggling to keep myself together at the minute. The pride to dig that deep at 21-0 down…
"There's a lot of tears being shed at the moment – tears of joy. We've got a cracking game tomorrow now in the semis.
“[New Zealand] are in great form and it's now about resetting and coming out firing tomorrow.”