The visitors took the lead through Jonathan Howson after 20 minutes, but Dublin equalised and Huckerby scored the winner. Huckerby also went close with two more shots late on. Neil Sullivan got his fingertips to the first and the second came back off a post.
The Whites kicked off, kicking towards the River End goal, but it was the Canaries who did most of the attacking in the early stages.
Jurgen Colin went close with a low shot and Dublin was only inches away from getting his head to newcomer Simon Lappin's free-kick. Chris Brown set up Huckerby who put the ball in the net but was flagged offside.
After 14 minutes Huckerby and the other City new boy, Mark Fotheringham, combined to set up Lee Croft but he blasted the ball over the bar.
Tresor Kandol produced the first shot for Leeds but it went wide. However, the visitors went ahead in the 20th minute.
Kandol headed on Sullivan's long free-kick, Robbie Blake pulled the ball back and Howson got between two City defenders and beat David Marshall low to his left with a firm shot from 10 yards.
Jonathon Douglas was booked for tripping the lively Huckerby after 32 minutes.
Leeds won a corner in the first minute of the second half, but it was cleared before it reached the six-yard area.
Dublin equalised in the 59th minute. Youssef Safri's cross from the left was cleared towards the edge of the Leeds penalty area and Dublin's right-foot volley found the top right-hand corner of the Leeds goal.
Andy Hughes replaced Colin as Norwich went in search of the winner. Leeds brought on Ian Moore for Kandol, while Norwich replaced Brown with Chris Martin.
With Leeds also looking to take the lead, Norwich broke out of defence through Fotheringham and Croft who released Huckerby down the left wing.
He cut inside and beat Sullivan low to his left with a firm, angled left-foot shot.
It turned out to be enough for Norwich to net a much-needed three points.