Darren Carter fired Cheltenham into a 62nd minute lead, turning the ball home after the Vale defence only half cleared a Chris Zebroski cross, but that lead lasted just five minutes before Louis Dodds levelled.
Substitute Daryl Duffy then missed a chance to bag all three points deep in injury time and the Robins boss said it would have been a deserved victory.
"I can't believe we missed that chance at the end," Yates said. "We just expected Daryl to stroke it into the empty net. Everyone was up on the bench and I thought it would have been just what we deserved.
"I thought we were excellent. They are a really good side - you can see that - they have some big strong lads and Pope has got 20-odd goals already .
"We kept going until the 93rd minute but just a bit shell-shocked that we have not won the game. We had numerous chances to win it.
"We were disappointed with the goal but more than happy with the performance against a side that has been there all season. We passed it really well and created lots of chances.
"We thoroughly deserved to win but you don't always get what you deserve. We have to make sure we go to Rochdale next week and get three points."