Still smarting from being dumped out of the FA Cup last time out by non-league Tamworth, Cheltenham went ahead in the 56th minute when Jamie Cureton finished Jason Taylor's cross.
It was a first league win in four attempts for the Robins and manager Yates insisted it was the least his side deserved.
"We owed everyone, we owed it to ourselves, our supporters, the club and it's the start of the way back," Yates said.
"There's 30 games to go and we've ticked one of them off. It was a good away performance and I thought we were the better side.
"They were obviously desperately disappointed from last week [the Tamworth defeat] but it shouldn't take that to put in a performance.
"They are a good bunch but we did not do ourselves any favours last week. Our league form over seven or eight games has been decent and hopefully we can come through late.
"They did not go out last week to look disinterested, it just did not happen. Unfortunately we got found out at Tamworth and took a lot of stick from it.
"We've answered a couple of questions back. We have quality and endeavour and the spirit has never been in question."
And Wycombe boss Gareth Ainsworth admitted he could have no complaints after seeing his side slip up in front of their own fans.
"It was a hangover from Tuesday, it was a lacklustre performance and not one we've come to expect at Wycombe," Ainsworth said.
"I didn't think we deserved to win the game at all. If we'd have drawn the game we would have been lucky even though their winner was a wicked deflection.
"But we have to get over it as quickly as possible.
"We felt it was unjust at Swindon and we probably carried that with us. But I will address it.
"We have played well in some home games, we were just lacklustre. They were there for the taking, they weren't impressive at all, but we gave them the points,
"I don't like that and it won't happen very often in the future - we need to make sure those nothing games go in our favour.
"It was hard that second goal. It did take them to score their first for us to kick in. We did create a few chances after that but there is no way we would have deserved it.
"We just about got what we deserved but at least we don't get hammered anymore."