2012 and 2016 bronze medallists Canada edged out Brazil 4-3 with goalkeeper Stephanie Labbe saving two spot-kicks, after the sides had remained deadlocked at 0-0 after extra-time.
The USA were also victorious on spot-kicks, defeating the Netherlands 4-2 after a thrilling 2-2 draw in normal time - Alyssa Naeher the hero for the Stars and Stripes.
Beckie completed 105 minutes - substituted in the additional 30 minutes - while Dahlkemper completed the full 120, on another pulsating day of action in the quarter-finals.
Canada became the first side to reach the last four in Japan, as the Group E and F runners-up locked horns.
Much of the pre-match discussion for the last eight encounter in Rifu focused on the excitement of legendary duo Christine Sinclair and Marta lining up against each other.
A head injury to Sinclair hampered the early exchanges at the Miyagi Stadium but it was Canada who fashioned the game's first real opportunity as Rebecca Quinn sliced over the bar.
At the other end, Tamires saw an effort suffer the same fate.
Right-winger Beckie almost carved an assist, heading into Jessie Fleming's path and then firing to the far post - only for both strikes to drag wide.
Labbe was then called into action, as Debinha looked to capitalise on a Vanessa Gilles but was thwarted by the Canada goalkeeper in the last major action of the half.
After the break, both sides pressed for a winner.
Andressinha tried her luck from range but her curling strike was comfortable for Labbe, before Gilles struck the crossbar with a header from a Beckie delivery.
In a late flurry of pressure from both sides, Labbe denied Debinha once more, while Deanne Rose was blocked and Alonso Costantino Angelina drilled wide.
Ludmila almost snatched an injury-time winner but could not clip the ball past Labbe and the sliding Gilles, and extra-time beckoned.
The additional 30 minutes yielded plenty of action but still no goals - Beckie, Adriana Leon, Marta and Debinha all firing off-target, before Labbe produced an excellent stop to tip Erika's late header wide.
As ever, there was plenty of drama in the penalty shoot-out. Sinclair opened the spot-kicks but saw her effort saved by Barbara, while the next six strikes were all successful.
Alves da Silva Andressa's attempt was then saved by Labbe, before Gilles turned the tie in Canada's favour, and Labbe's block of Rafaelle's effort clinched the victory.
Meanwhile, the United States booked their place in the last four with victory in Yokohama, as the 2019 World Cup finalists locked horns.
Former City star Sam Mewis was on the scoresheet, as the World Champions contested an entertaining 2-2 draw with the Netherlands.
Vivianne Miedema had opened the scoring for the Olympic debutants on her 100th cap with her ninth goal of the tournament – an excellent turn and finish on 18 minutes.
The USA responded well to the setback with two goals in quick succession, levelling through Mewis’ diving header before Lynn Williams, who provided the assist, then turned the game on its head with a well-taken half-volley three minutes later.
Miedema pulled the Netherlands back on level terms nine minutes after the restart, cutting inside and drilling into the near post.
Christen Press thought she had restored the USA’s lead – only for the offside flag to be raised – while Lieke Martens spurned a glorious chance for a late winner from the spot at the other end, placing a penalty straight at Naeher after Kelley O’Hara had upended Lineth Beerensteyn.
Jill Roord, Danielle van de Donk and Beerensteyn all fired wide in extra-time, and Martens saw a header disallowed for offside after the ball had rebounded off the crossbar.
Miedema almost bagged a hat-trick but was denied by Naeher while Megan Rapinoe drove into the side-netting and Press headed over.
Press and Alex Morgan both found the net late on but both goals were disallowed for offside, and so the game went to penalties.
Miedema saw her effort saved, while the USA - including former City midfielder Rose Lavelle - were all successful.
Aniek Nouwen was then also denied from the spot by Naeher and Rapinoe stepped up to send the USA through to the semis, where they will face Canada in Kashima on Monday 2 August at 9:00 UK time.
Australia will face 2016 silver medallists Sweden (featuring former City player Kosovare Asllani) in the other semi-final at noon UK time the same day.