The 28-year-old, on the transfer list and the subject of at least two bids by Championship rivals Coventry, broke the deadlock with a 48th-minute header from Ian Henderson's cross.
The former Peterborough forward nearly scored on several other occasions and worked hard throughout. The only blot on McKenzie's night was a second-half booking for kicking the ball away.
City's place in the second round was clinched by Nigerian midfielder Dickson Etuhu, who swivelled in a crowded goalmouth to score with a left-foot shot in the 64th minute.
Canaries manager Nigel Worthington fielded only five first-team regulars in a much-changed line-up, leaving several of his star players back in East Anglia.
But Norwich were still more than good enough to win and they seldom had to move out of second gear, despite Torquay goalkeeper Nathan Abbey making a series of good saves in both halves.
City even survived the loss of right-back Andrew Cave-Brown, carried off with an ankle injury midway through the first half.
Despite the industry of Mickey Evans up front, Torquay seldom threatened and Norwich keeper Paul Gallacher's only worrying moment was a 25-yard free-kick which he smothered from Jamie Ward in the second half.