Tomas Rosicky scored after just 72 seconds to put Arsenal ahead and Arsene Wenger's men managed to keep hold of their slender advantage, despite heavy pressure from Spurs.
Rosicky's goal proved to be the decisive moment at White Hart Lane, but it was a harsh result on the hosts, who had chances through former Arsenal forward Emmanuel Adebayor in the first half and dominated after the break.
Arsenal moved within four points of leaders Chelsea at the top of the Premier League after claiming the local bragging rights in north London, while Tottenham remain in fifth, seven points behind fourth-placed Manchester City.
However, Sherwood was encouraged by his team's response to Arsenal's quick start and insisted they were the better side.
He said: "Especially after going behind so early in the game, you fear the worst.
"But they did me proud and stuck together, and took the game to Arsenal, we're talking about a team who are pushing for the title here and we didn't deserve to lose that game and I think we deserved to win it.
"I'm a winner and I want to win. I'd rather play badly and win.
"But they had one shot in the second half with Per Mertesacker and Hugo (Lloris) made a great save.
"They came here to counter attack and that's credit to us.
"They played really well in the first half but the boys stuck to their guns, showed some quality and we should have scored, simple as that."
