The hosts had lost their previous four games but there was little to suggest a crisis of confidence as Bobby Grant and Ashley Grimes gave them a 2-0 lead with two classy strikes at the end of the first half.
Goalkeeper Scott Brown was far and away Cheltenham's best player as he prevented Dale running away with the result in the opening 45 minutes.
The Robins' hopes of a comeback were briefly raised by a Phil Edwards own goal three minutes into the second period, but further strikes from Grimes and Dele Adebola ended Dale's losing sequence and scuppered Yates' hopes of moving up to second place in the League Two standings.
"That was absolutely shocking, we were lucky to get away with conceding four because it could have been 15," Yates said. "We had a chance here to cement our place in the top three, and we will probably be out of it by the end of the weekend.
"The fans who travelled deserved better than that, and I deserved better. That has really upset me, it was an absolute disgrace.
"Scott Brown is the only player to come out of that game with any credit, he was outstanding. The rest need to take a look at themselves in the mirror. Browny can have a day off - for the rest, Christmas is cancelled.
"That was not acceptable. We defended like pansies. There is a long way to go and we'll be alright. But if they think they will be off tomorrow and on Christmas Day they can think again. I've been as good as gold with the players, we look after them, but I'm not having that."
It was the first win in five games for Rochdale, leaving Spotland boss John Coleman delighted, and he said: "That was much more like it, though we made that many chances against Exeter recently but lost. The difference was, on the night, the ball went into the net for us and when that happens it gives you a feel-good factor.
"When we conceded a stupid goal we could have easily shipped more because of the mode we have been in of late, but the lads had a steely determination about them and were determined to get the next goal."