A flying start from the Cobblers put them two goals to the good and left Cheltenham manager Mark Yates thinking he had 'picked the wrong team.'
The visitors, however, gradually found a foothold in the game and proved Yates' tactical changes to be the correct ones.
"I wanted three combative midfield players and I wanted to get them to protect the two centre-backs, which I thought they did superbly well," he said.
"We could have got a bit of a tanking but I think we got what we deserved in the end."
Boothroyd, meanwhile, knew that his side had missed a great opportunity to keep in touch with the play-off places.
"As a group we've improved enormously but to lose three, win three and lose three isn't good enough," he said.
"So we're either naive or daft because we keep giving ourselves a mountain to climb.
"I'm very disappointed but we've got to make sure we stay in the pack now. We got exactly what we deserved because we didn't maintain our momentum for the rest of the game after such a good start."
Source: Cobblers Mad