The Iceland international attacking midfielder was one of two changes to the Spurs team that lost at Arsenal last time out, and made the most of his recall by netting in each half.
The other introduction made by Andre Villas-Boas - new £11.5million signing Christian Eriksen - claimed an assist on what was an impressive debut.
Norwich, who were unchanged, were on the back foot early on at White Hart Lane and almost fell behind after 10 minutes.
Eriksen ran into the area on the left, nutmegged Michael Turner and fired in a low shot on goal. John Ruddy failed to hold the ball and the lurking Roberto Soldado then improvised - only to see his backheel hit the post.
However, on 28 minutes, Spurs did find the net as they registered their first goal from open play this season. Soldado chested Danny Rose's ball from the left down to Eriksen on the edge of the area. The Dane played a sweet, short pass into the area for Sigurdsson to run onto and he fired across Ruddy into the corner from 10 yards.
Ricky van Wolfswinkel spurned Norwich's best chance of the first half, with Rose blocking his effort after a Nathan Redmond left-wing cross.
Spurs enjoyed 73% possession in the first half, and only needed four minutes after the restart to double their advantage.
Eriksen put Paulinho free on the right edge of the area and the Brazilian had time to pick a low cross. The ball appeared to go between the hands of the diving Ruddy and that allowed Sigurdsson to tap home his second of the afternoon at the far post.
The same player stung Ruddy's palms with a 30-yard drive as he hunted a hat-trick and the energetic Andros Townsend and Rose also went close to making it 3-0.
Van Wolfswinkel had another sight of goal in the 71st minute but took too long over his shot, allowing Rose to block again.
Erik Lamela was brought on by Villas-Boas for the closing stages but there were to be no further goals for Spurs on an afternoon when they had 23 shots at the Norwich goal, with seven of those on target and six blocked.
There was, however, still time for a late scare for Hugo Lloris, who was lucky to escape with just a yellow card when he punched the ball marginally outside his own area with five minutes left.
The result moves Spurs up to second in the Premier League table after four games - behind new leaders Arsenal, but above Liverpool on goal difference. Norwich slip to 13th.
Tottenham host Tromso in their first Europa League Group K fixture on Thursday night, before travelling to Cardiff in the Premier League next Sunday. Norwich meet Aston Villa at Carrow Road next Saturday.