Russell Penn scored the only goal of the game after just five minutes with a low drive from the edge of the penalty area - ending Exeter's hopes of making the play-offs.
Cheltenham must now beat Bradford and hope Rotherham slip up in their home encounter against an Aldershot side clinging to their Football League life.
"At this stage of the season against a good side, we have done a really professional job," Yates said. "I have asked the boys to take it to next Saturday and that is what we have done.
"All we can do is take care of our result and see what happens. Hopefully get a big crowd next Saturday at home to Bradford and you never know what is around the corner. Aldershot have got something to play for now which we are happy with. We have got to take care of ourselves.
"If results don't go for us we can hopefully go into the play-offs in a positive frame of mind. It was a big win for us.
"We will worry about what happens next week but if we don't get there, there will be a moments disappointment - and we are allowed that. Then we will be focused purely on the play-offs. But right now it is a big well done to the whole squad and our fantastic support."