With Switzerland trailing 1-0 courtesy of Jakub Blaszczykowski's 39th-minute strike, Shaqiri produced a brilliantly executed bicycle kick to stun Poland and force an additional 30 minutes.
However, if Shaqiri's strike is now the standout contender for goal of the tournament, team-mate Xhaka's missed penalty was at the opposite end of the quality spectrum, the new Arsenal signing smashing his strike well wide of the target with Switzerland's second effort to leave them playing catch-up.
Grzergorz Krychowiak made no mistake with his chance to wrap up the victory, burying his shot beyond Yann Sommer and sending Poland into the last eight for the first time.
Poland dominated early on Saint-Etienne but the pressure had been building prior to the 82nd-minute equaliser – Ricardo Rodriguez drawing a fantastic save from Lukasz Fabianski with a free kick, while Haris Seferovic rattled the crossbar.
The anthems had barely finished when a Swiss defensive mix-up led to a chance for Robert Lewandowski.
Sommer's throw was not dealt with well by Johan Djourou, forcing the goalkeeper to rush out and deny Lewandowski a shot at goal, before Arkadiusz Milik lifted over the crossbar of an unguarded net.
Poland continued to look most likely to break the deadlock, with Krychowiak heading over unmarked and Kamil Grosicki miscuing his shot, and Milik skied an effort from close range following a quick counter-attack.
A rare chance fell Switzerland's way in the 35th minute but Fabian Schar could only head straight at Fabianski from a corner, while Blerim Dzemaili forced the keeper to tip over his crossbar with a deflected strike soon afterwards.
With the Swiss beginning to settle, Blaszczykowski's opener was well timed. A Grosicki pass found the Borussia Dortmund midfielder in plenty of space on the right-hand side of the penalty area and he had plenty of time to pick a shot between Sommer's legs to give Poland a deserved lead.
The goal was Blaszczykowski's second in as many matches and he almost added further to his tally after cutting inside and unleashing a shot from 20 yards eight minutes into the second half, with Sommer able to push the ball out for a corner.
Schar escaped with a yellow card following a late and heavy challenge on Lewandowski as the match became fractious, but Switzerland did show signs of life in attack following the introduction of youngster Breel Embolo in place of Dzemaili.
Fabianski showed great strength to beat Embolo to a cross and cling on to the ball in the 67th minute, before launching himself across goal to superbly keep out a Rodriguez free-kick that looked destined for the top corner.
The pressure was mounting and Seferovic hit the crossbar from 12 yards as Poland failed to clear their lines, before Shaqiri made the breakthrough in truly spectacular fashion. A cross from the left bounced out to the edge of the box, where the Stoke City winger positioned himself to send an overhead kick hurtling into the bottom-right corner.
While the opening period of extra time passed without incident, the momentum was firmly behind Switzerland and Poland were forced on the back foot, with Fabianski producing another fantastic save to deny an Eren Derdiyok header after the break.
Chances to prevent penalties came and went for both sides in the closing stages of extra time but it was left to penalties to settle the tie and it was Xhaka who became the villain for Switzerland as Poland proved perfect from 12 yards.