The Robins are, however, still very much in the tie as they prepare to head home for the return date on Sunday.
A solitary effort from Roy O'Donovan proved to be enough for the Cobblers on Thursday, and they now have one foot at Wembley.
They can expect to face a much improved Cheltenham side at the weekend, though, with Yates relieved to be in with a shout of securing promotion out of League Two.
He told Sky Sports: "We are losing. They were miles better than us tonight.
"It's 1-0 and I'm probably the happiest man in the ground right now.
"They didn't create too many chances - sorry, efforts on goal that the keeper has had to make saves from. We defended heroically but we didn't play anywhere near what we're capable of.
"We had some chances, but I'm happy that it's 1-0."