The first half was a drab affair as both team struggled to adapt on a bobbly pitch battered by a swirling wind.
However, the difference was the set-piece skills of Darren Currie, as he sent over corners on 52 and 86 minutes which were finished off by Garvan and Fuller respectively.
Leicester had made a lively start when Iain Hume had two decent chances, tamely shooting straight at Shane Supple with the first, and then firing a firm low drive narrowly wide after Danny Tiatto's long ball went over Scott Barron's head.
Town first threatened on 11 minutes when Leicester keeper Paul Henderson fumbled a Currie free-kick and a goalmouth scramble ensued before the visitors scraped the ball away.
Fabian Wilnis then saw a deflected shot loop into the arms of Henderson, before Nils-Eric Johansson headed wide when left unmarked to meet a Joey Gudjonsson corner.
Fuller claimed a penalty when he went down in the box on 38 minutes, but ref Paul Taylor was unimpressed and it did appear as if he had simply stumbled.
Town's best first-half chance came a minute before the break when Richard Naylor headed a Currie corner over from six yards at full-stretch.
Garvan sliced an early shot wide in the second period, but almost scored with his next effort, a firm low shot after Fuller teed him up, only for Henderson to make a fine diving save.
Alan Maybury had a 25-yarder palmed away by Supple, before Town broke from the corner and Fuller played in Currie, who came inside and fired low, with Henderson turning it around the post.
Ipswich scored from Currie's set-piece, which Jason De Vos helped on and Garvan steamed in to smash home from close range.
Leicester almost hit back just before the hour when a clever back-heel from Hume freed strike partner Matty Fryatt, who shot across the face of goal from 20 yards.
A clever clipped ball by Currie was almost helped on by Westlake before Fuller volleyed well wide after a nice 1-2 with Garvan.
While it stayed 1-0, there was always a chance for Leicester, although the visitors had carried little threat since the early stages, and Town made sure there would be no come back with four minutes remaining.
Sub Dean McDonald earned a corner after his shot was deflected wide and Currie lifted the ball in for Fuller to escape his marker and power a six-yard unmarked header into the corner.
Leicester threatened in stoppage time as a Gudjonsson corner dropped to Johansson, whose well-struck volley was turned around the post by Supple at full-stretch and then sub Elvis Hammond saw firm drives charged down by Scott Barron and De Vos.
The win extended to seven Town's streak of successive home wins over a Leicester side whose last win at Portman Road came way back in 1988, and they rarely looked like ending that run in this match.