After their victory at BC Place in Vancouver on Thursday (Friday AEST), Switzerland will now face the winners of Friday's clash between Colombia and Ghana in a bid to reach the quarter-finals for the first time since their home tournament in 1954.
The Swiss took the lead in the 10th minute through Breel Embolo, but Johan Manzambi did all the hard work.
The Freiburg midfielder sprinted to the box and beat the defender before keeping the ball within the lines and cutting back to Embolo to turn in from close range.
After the break, Switzerland needed only 48 seconds to find the net again, and with some very welcome help from the opponents.
Algeria twice failed to clear a ball in the box after a cross from Denis Zakaria. The ball eventually found Dan Ndoye, who steered a low shot into the bottom corner.
Fabian Rieder had the chance to add a third in the 81st minute, but he somehow failed to finish from close range and with keeper Luca Zidane spread out on the turf.
Rieder sneaked behind the Algeria defence, but his effort bounced into Zidane's arms before being cleared away.
Switzerland had not won a knockout-round match since 1938 in France, but in 1954 they claimed a playoff game to reach the quarter-finals.
Although the Swiss had reached the round of 16 at the past three World Cups, there was a smaller field of 32 teams in each of those tournaments, and they could get no further.
Algeria were eliminated after returning to the World Cup for the first time since 2014, when they advanced to the knockout round but fell to eventual champions Germany.
- with AP
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