After a scrappy start from both sides, Preston had the first real shot on goal but Richard Cresswell's drive from the edge of the penalty area was comfortably saved by Robert Green.
Michael Keane saw a 30-yard drive fly well over the bar in the 31st minute, before North End almost took the lead two minutes before half-time but David Healy's drive was superbly tipped around the post by Green. From the resulting corner the ball eventually found it's way to Iain Anderson but his shot from the edge of the box was cleared off the line by Mulryne.
City came out much improved in the second half, after not having registered a single sht on target in the first, and looked set to go in front in the 54th minute when a David Nielsen cross was met by Paul McVeigh, but his shot was well blocked by Alexander.
Moments later Mulryne saw a cross-cum-shot flash across the face of the goal, before the home side took the lead just after the hour mark when Roberts glanced home a superb header after climbing above Sean Gregan to meet Steen Nedergaard's inswinging free-kick.
The home side then increased their lead in rather unfortunate circumstances when a back-pass from Alexander was completely misjudged by Preston keeper Moilanen and the ball rolled under his foot and into the back of the net, musch to the City fans' delight.
Norwich completed the scoring two minutes before the end when boo-boys' target Phil Mulryne's curling free-kick from 25-yards out beat Moilanen at his near post.
Preston almost pulled a late goal back but a Healy lob was turned away by Green. Following this much-needed win, Norwich are now seventh in the league - but equal 6th on points, and six points ahead of our nearest rivals. However we stand to lose ground on the top 6 after Saturday's games, but we'll still have a game in-hand.