Belgium, tipped by many as outsiders to challenge for the FIFA World Cup, dominated proceedings in their tournament opener in Belo Horizonte, but found themselves behind after 25 minutes through a Sofiane Feghouli penalty.
Wilmots made a trio of substitutions by the 65th minute and the triple change worked to perfection as first Marouane Fellaini and then Dries Mertens found the net in the space of 10 second-half minutes.
The Belgium coach conceded that his side had not performed well enough in the first half of the Group H clash and was pleased with the patience showed to secure the victory.
"I am relieved, yes," he quoted as saying by Het Nieuwsblad. "We knew it would be difficult.
"Algeria took to the field well, while we played too slow.
"I seek no excuses though, it turned out the way I wanted.
"We had 67 per cent of possession and gave nothing away. Also, that penalty was not a chance, that was a minor mistake. Eventually we remained patient and waited until Algeria became tired.
"I'm really happy. We wanted the three points, we have the three points."
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