Breel Embolo, who was born in the West African country, scored the only goal when he stroked home Xherdan Shaqiri's low cross.
The 25-year-old's celebrations were understandably muted, but his goal was nevertheless vital for Switzerland, who will be aiming to at least match their quarter-final appearance at Euro 2020 last summer.
Cameroon had the upper hand in the first half but Switzerland started the second half well, with Embolo's goal coming in the 48th minute.
Switzerland coach Yakin said: "Of course, you need patience. It was a mature performance.
"Of course there is some uncertainty in a World Cup, not everything can go well from the start, but I'm happy with the possession, with how we got it converted.
"Maybe 1-0 or 2-0 is not sufficient for my taste, maybe we could have scored more goals, but all in all we had a mature performance by our team."
Asked about Embolo's reaction to scoring, Yakin said: "We know his story, his past in Cameroon, and Breel, you can be friendly up to the kick-off and then afterwards they are going to be your opponents and we need you in our team.
"You've got an opponent, of course he wants to score a goal for his team and he fulfilled his task, so we are really happy about this.
"He also had very good moments during the match, his dynamics and his movements. Breel fulfilled his task well."
Cameroon coach Rigobert Song said his team had been punished for not being clinical enough.
Karl Toko Ekambi and Eric-Maxim Choupo-Moting were guilty of wasting good openings in a first half Cameroon had the better of.
"Generally in football, when you're dominating you need to hit the net," Song said.
"Yes, we had more opportunities than they did (in the first half) but we didn't have that finishing touch. That was what made the difference.
"We showed we were hungry for that win but the goal simply didn't come. We are playing high-level football here, it's no walk in the park. You can monopolise ball possession, it's taking chances that counts."
The defeat meant Cameroon have now lost eight consecutive World Cup finals matches, and face a tough route to the knockout phase with matches against Serbia and Brazil to come.
"We haven't lost hope, we have still got two games to play," former Liverpool and West Ham defender Song said.
"It's all systems go now. We have still got that faint hope. I believe the next game will be different."