Patricio dived full length to block Jakub Blaszczykowski's penalty, Poland's fourth, allowing substitute Quaresma, whose goal late in extra-time beat Croatia in the round of 16, to bang in Portugal's fifth in the shootout.
Two hours earlier Robert Lewandowski put Poland ahead with a second minute strike only for teenager Sanches to equalise with a superb shot 33 minutes into his first international start.
Precious little happened from then on but Portugal, beaten on penalties by Spain in the semi-finals four years ago, will not care as they advance to the last four at the Euros for the fourth time in five tournaments to face Belgium or Wales.
"(The goal) was an amazing moment. I'm happy to be man of the match again, but the individual comes after the team," said Sanches, who also scored the second shootout penalty.
Cristiano Ronaldo, who four years ago did not get to take his planned spot kick when he was fifth in line as Portugal lost to eventual winners Spain, went first this time.
Joao Moutinho and Nani also scored confidently from the spot before Patricio swung things Portugal's way with his fine save and Quaresma applied the coup de grace.
Poland had converted all five of their penalties in knocking out Switzerland in the last round, also after a 1-1 draw, but could not repeat the feat.
"The players put a lot of heart into this match and played bravely throughout this tournament and this defeat hurts," said Poland coach Adam Nawalka.
"We haven't lost a match at the tournament - we were knocked out on penalties. We've moved forward - this tournament has proved that. We played some very good football but this defeat hurts."
DEFENSIVE MISJUDGEMENT
It was a somewhat strange encounter, with an all-action first half followed by 75 minutes of inaction.
Poland went ahead when Portugal defender Cedric’s misjudgement allowed a long ball to bounce over his head to Kamil Grosicki. He drove on down the left and squared for Lewandowski to guide the ball in.
It was the striker's first goal of the tournament and, at one minute 40 seconds, the second-fastest in European Championship history after Dmitri Kirichenko's for Russia against Greece after 65 seconds in 2004.
Poland looked the more confident team for the next 20 minutes but failed to press home their advantage as Portugal settled and began to make inroads.
The Portuguese levelled when Sanches, due to join Lewandowski at Bayern Munich after the Euros, played a neat 1-2 with Nani and found time to shift the ball onto his left foot at the edge of the box and smash it past Lukasz Fabianski.
Both sides came out for the second half seemingly under instruction to slow things down and the life bled from the game.
The only clear chance of the half came in the 85th when substitute Moutinho lobbed a clever pass for Ronaldo who, having taken the time to glance at the keeper’s positioning, failed to make any contact on the dropping ball with the goal gaping.
Little changed in the additional 30 minutes, with the near-64,000 crowd drugged into a drowsy-near silence, seemingly knowing what was afoot almost from the restart.
Portugal have failed to win any of their five games here in 90 minutes, having drawn three group matches and beaten Croatia in extra time but midfielder Sanches, on the greatest day of his fledgling career, was in no mood to apologise.
"It takes a lot of hard work to be this lucky," he said. "People criticise us but we don't care because in the end we are in the semis."
(Editing by Ken Ferris)
