After losing 1-0 at Sixfields on Thursday night, Marlon Pack had the chance to level the tie in the 14th minute, but Lee Nicholls produced a fine save to keep out the midfielder's penalty.
Luke Guttridge's wonder strike doubled the Cobblers advantage and, after the Robins missed a host of chances, they held on to set up a final showdown with Bradford.
Yates said: "If we would have played all night we still wouldn't have scored but the town can be very proud of its players.
"Their goal was a great strike and Scott had no chance with it. We were the better team on the night and they were the better team last Thursday and unfortunately things haven't gone for us.
"Marlon is disappointed in the dressing room because he thinks he has let people down - there are tears pouring down his face. But Marlon Pack has been magnificent for the past three seasons at this football club.
"Tonight he stood up in front of a packed house and was brave enough to take a penalty. There were plenty of others who wanted it but he did it as he has all season and unfortunately the keeper's guessed the right way.
"The boys have to pick themselves up and have a good night out in Cheltenham. They have been fantastic all season. Our resources don't get near a lot of clubs in this division but that just shows how decent we are."