The Iceland international netted one either side of half-time, but the score could have reached double figures such was the hosts' control of the game.
Tottenham created chances galore, with new signings Christian Eriksen and Paulinho particularly impressive and Norwich had Sebastian Bassong and John Ruddy to thank for keeping the scoreline respectable.
Eriksen then had a good chance, jinking inside from the right and shooting low to draw a good save from Ruddy. Roberto Soldado's audacious back-heel from the rebound was saved by a combination of Bassong and the post.
And ex-Tottenham defender Bassong again came to Norwich's rescue minutes later. Moussa Dembele took out two defenders with a feint and cut-back to Soldado, but the Cameroon international blocked his goal-bound effort.
A Tottenham goal looked inevitable, and so it proved in the 28th minute when Eriksen laid the ball into Sigurdsson's path as he burst into the area and the former Hoffenheim man finished coolly past Ruddy.
Norwich had not been in the game, but they had a half-chance 34 minutes in through Ricky van Wolfswinkel. The Dutchman met Nathan Redmond's left-wing cross but was thwarted by a last-ditch Danny Rose tackle.
Tottenham picked up where they left off after half-time and were soon 2-0 up. Paulinho’s low cross from the right evaded Ruddy’s dive, presenting Sigurdsson with an open goal.
Norwich were buckling under concerted Tottenham pressure and they had Michael Turner to thank for not conceding a third, as he made a fantastic intervention to deflect Townsend’s right-wing cross over the bar, with Soldado lurking behind him.
Chris Hughton’s side finally had a chance of their own on 68 minutes, the ball breaking to Van Wolfswinkel in the area, but Rose foiled the former Sporting Lisbon striker with a good block.
There was controversy late on as Hugo Lloris left his area to make a punched clearance. The Norwich fans called for a red card but referee Lee Mason was lenient, only showing a yellow to the France international.
