The Desert Foxes took a surprise lead through Sofiane Feghouli's first-half penalty, but the Belgians fought back and had Marouane Fellaini and Dries Mertens to thank after both came off the bench to score and scrape a win for their side.
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"The players were very tense and stressed," the 45-year-old said after the game.
"The circumstances didn't help them either but ultimately we managed to be patient before capitalising on the Algerians' tiredness. During half-time, I gave the players the message that the bench will make the difference and that turned out to be true."
Belgium, who have not featured in a World Cup finals since 2002, struggled to make an impact in large portions of the game against their less-fancied opponents, but Wilmots insisted he was impressed by his team's attitude.
"It was a difficult match. We made a very little mistake, but at World Cups they are punished," he continued.
"In the second half, this young team showed mental strength."
Belgium now face Russia on June 22 and South Korea on June 26 to complete their Group H campaign.