Murat Yakin's side broke the deadlock in the 10th minute when Johan Manzambi burst forward down the left, ran to the byline and picked out Embolo, who turned the ball in from close range. Switzerland doubled their lead 48 seconds into the second half as Dan Ndoye swept home after Algeria defender Rafik Belghali's clearance fell straight to him, with Denis Zakaria heavily involved in the build-up.
Substitute Fabian Rieder somehow failed to add a third from close range in the second half, but it made no difference to the outcome. The result marked Switzerland's first World Cup knockout victory since 1938, and sends them into a last-16 meeting with the winner of Colombia against Ghana, back in Vancouver.
Embolo said the result was built on a team performance rather than individual quality.
"The most important thing is the victory," the Switzerland forward said. "I spoke about being clinical. We suffered at the start of the game, but managed to score a superb team goal.
"I'm at the end of the team effort. They started strongly, but congratulations to our team for controlling the match and especially with how we came out at the start of the second half. Algeria is a strong team and we're happy to win with a clean sheet too."
Algeria's Riyad Mahrez reflected on his side's exit from the tournament, admitting the margins had been too fine.
"Our aim was to go through and I think it was a game we could have won," Mahrez said. "But we conceded twice on mistakes and at this level, you can't get away with it. There are always positives to take away from matches: we did manage to get out of the group stage, but we conceded too many goals to aspire to more."
Mahrez also confirmed the game marked the end of his international career.
"This was my last match with Algeria," he said.
Algeria, managed by former Switzerland head coach Vladimir Petkovic, finished their World Cup campaign having conceded nine goals in four matches, a defensive record that ultimately caught up with them in the last 32. Ibrahim Maza had their best chance of the match, dragging a shot wide in first-half stoppage time, but they otherwise struggled to create clear openings against a well-organised Swiss side.
Mahrez's assessment carried the weight of a player closing out his Algeria career at a World Cup, having helped his country reach the tournament's knockout stages before their run ended against Switzerland in Vancouver.