On-loan teenage defender Luke McCullough came off the bench just before the break for his debut and was joined by winger Kaid Mohamed at the start of the second half.
The hosts had gone behind to a Danny Rose strike after 23 minutes, but Mohamed's arrival sparked Cheltenham's revival and Paul Benson looped a header over Shots keeper Jamie Young midway through the second half for the equaliser.
The visitors were then forced to hold on with defender Anthony Tonkin picking up a second yellow card 17 minutes from time.
"I thought all the substitutes had a really good impact on the game," Yates said.
"The players showed tremendous character and we played some really good stuff in the second half when we went back to two widemen - maybe I should have done that from the start. I stuck with the same team and maybe, looking back, I should have gone with some more width knowing they were going to flood the midfield. We were a bit narrow in the first half.
"We knew Luke McCullough could handle the occasion. He has been unlucky in that we have taken him and then have got Michael Hector in. But he has shown what he is about as a defender and has a good strike on him and he should be pleased with his league debut.
"Obviously I'm disappointed that we haven't won the game. But it is another game unbeaten and it could be a good point."