However, Dario Gradi's players contributed to an entertaining game and embarrassed City in the 19th minute when Luke Varney gave the visitors the lead.
Leon McKenzie equalised five minutes into the second half but that was as good as it got for the home fans.
Norwich kicked off attacking the Barclay End goal and forced two corners in the first 10 minutes without testing Ben Williams in the Crewe goal.
In the 14th minute Dean Ashton blazed his shot over when he should have hit the target.
Varney's header was on target a minute later but Robert Green reacted brilliantly to turn the ball over the bar.
Unfortunately for Green in the 19th minute, Eddie Johnson got the better of Craig Fleming and released Varney with a low through ball and this time Green's right hand failed to keep out his fierce left-foot shot from the edge of the area.
As Norwich battled to get back on terms, Ashton's shot was deflected for a corner.
Dean Marney slammed a free-kick against Williams' left-hand post and two minutes before half time McKenzie should have scored inside the same post when he got a foot to Ashton's right-wing cross but his shot hit the bar and skidded over.
Five minutes into the second half McKenzie made amends for that miss when he wriggled free of his marker and beat Williams with a low left-foot angled shot from eight yards.
Ashton's shot on the turn was then blocked by a defender and his free-kick hit one of his own players.
Youssef Safri replaced Jason Jarrett after 63 minutes and had a shot saved within seconds of coming on.
Darren Huckerby's shot was blocked and Ashton fired wide from long range when he could have taken the ball closer.