Ipswich had the better of the early opportunities, with Daryl Murphy sending a shot well wide before a heavy touch from the striker put the ball behind for a goal kick.
The visitors started to look more threatening as Kyle Lafferty tried to set up Steven Whittaker, before Nathan Redmond picked out Grabban, but Dean Gerken was on hand to keep the ball out of the back of the net.
The Canaries did break the deadlock shortly after, as confusion among the Ipswich defenders and Gerken allowed Grabban to make a move at the near post, heading in from close range following a corner.
David McGoldrick tried to get the home side back on level terms at the start of the second half, sending in a low through ball for Murphy, but Norwich managed to clear the danger.
Ipswich striker Murphy continued to get himself into dangerous positions up front, and had another opportunity to find the target shortly after, but sliced just wide from close range.
The visitors responded to Ipswich's bright start to the half, but Grabban hurried his effort, firing well wide while looking for his second of the afternoon.
The striker continued to provide a goal threat for the Canaries, forcing a save from Gerken at full stretch just short of the hour mark before trying to set up Bradley Johnson a couple of minutes later, only for the Ipswich keeper to make the stop again.
The home side pushed hard for an equaliser int he final 10 minutes, but Norwich got numbers behind the ball to make sure that they protected their slender advantage.
Redmond came close to putting the win beyond doubt in stoppage time as he tried to squeeze the ball in from a tight angle, but Gerken was alert to the threat.
The result moves Norwich to the top of the Championship table, having played a game more than those on seven points, while Ipswich sit 15th.