Nikica Jelavic, a Leighton Baines penalty and Leon Osman's rifled third had the visitors cruising after the break, but the League Two side swiftly hit back with a smart finish by Russell Penn to give them a glimmer of hope.
Seamus Coleman and Marouane Fellaini settled any nerves for the Premier League side as they completed a 5-1 rout and set up a fourth-round clash with Bolton or Sunderland.
"Two goals took a bit out of our sails," Yates said. "I was proud of our efforts. We responded well and tried to play. Everton were too strong a team and were too good for us.
"Everton picked a strong a team and that was a compliment. It showed the class they have. They gave us respect and the competition some respect.
"We played some decent football without getting behind them. We started to after the interval but then they scored. We just shot ourselves in the foot with the timing of the goals.
"There are things we could have done better but our commitment and work-rate was excellent. We can now concentrate on a 21-game season where we can cement our place in the top three and get promotion."
Penn was the pick of the Cheltenham players and his boss was relieved to see him finally get on the scoresheet.
"I'm delighted for him," Yates added. "One thing he's been lacking is a goal. Hopefully that can kick start his season."