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. Obviously I'm disappointed that we haven't won the game. But it is another game unbeaten and it could be a good point."
Aldershot manager Dean Holdsworth said: "It's a tough place to come. Cheltenham are a good team with some good players, so we have to be pleased with a point.
"It showed that we have some character and some resilience in the side. It was a big test and they have put in a massive shift. I told the players to make it a big week with a home game against Torquay coming up which will be our next cup final."
Tonkin's dismissal followed two fouls on former Aldershot winger Jermaine McGlashan. "The thing is I know Jermaine very well and I know what he does," Holdsworth added.
"There was nothing Anthony could do - he just stood there. The problem is that the referee has not played the game. Jermaine wins you a lot of fouls and that's the game. We're grateful that we've not suffered too much."
- Sports & Recreation
- Anthony Tonkin
- Kaid Mohamed