Norris fired home a volley for Ipswich in the 11th minute of extra time after a tired Crewe failed to clear a corner.
The League Two side knocked out Derby County in round one and had the better chances to win the game before running out of steam.
Clayton Donaldson nearly added to his brace in the Railwaymen's 7-0 demolition of Barnet on Saturday but narrowly failed to connect to an 11th minute Joel Grant cross from seven yards out.
Roy Keane threw 31-year-old Trinidad striker Jason Scotland on at half-time after his recent move from Wigan.
But former Swansea striker Scotland appeared to lack sharpness after his spell in the Premier league which yielded just one goal in 32 appearances.
Keane had reserve goalkeeper Brian Murphy to thank after saves from Shaun Miller with the last kick of normal time and from substitute Byron Moore at the start of extra time before Norris sealed the win.