With the wind howling as it had been all day, Britain withstood a first-half onslaught from Argentina when they were against the elements, at one point holding a ball off the line to prevent a try.
With the wind at their backs in the second half Britain had a string of chances, but Argentina's defence proved just as good - and they even had a chance to win the match as they took a shot at goal with the clock in the red.
The attempt missed, sending the game to sudden death extra time, and Britain had the first chance to clinch it, with a long-range penalty into the breeze that came back against the post.
Britain weren't to be denied, however: they won the ball from the resulting scrum, and Dan Bibby stormed through the tiring Argentine defence to score the try that put his team into the last four.
Rugby powerhouse New Zealand had no such luck as they were knocked out of the inaugural Olympic sevens tournament in the quarter-finals after a 12-7 loss to top seeds Fiji.
The All Blacks suffered defeats by both Japan and Britain in the pool stage, but were gifted a tournament lifeline when the Fijians beat the United States earlier on day two.
The 12-times world sevens series champions were unable to take advantage of their reprieve, however, losing in a breathless encounter at a wet Deodoro Stadium.
Fiji skipper Osea Kolinisau finished a brilliant length-of-the-field try to open the game many thought would take place in the final and Jerry Tuwai skipped through the All Blacks defence to add a second for the Pacific islanders.
Gillies Kaka had replied for New Zealand off a chip and chase while his team mate Rieko Ioane was in the sin-bin but the Fijians were able to hold on at the end to book a semi-final against Japan, who beat France 12-7.
2016 Rio Olympics - Rugby - Men's Quarter-final - Fiji v New Zealand - Deodoro Stadium - Rio de Janeiro, Brazil - 10/08/2016. Rieko Ioane (NZL) of New Zealand reacts after loss to Fiji.
Image credit: Reuters
The match was a tense affair decided in the closing moments as Japan scored with the clock in the red - and the joy on the faces of the players was clear to see.
The semi-finals, bronze medal match and final all take place on Thursday.