The visitors claimed an early lead through a Logovii Mulipola try, but Tommy Bowe got Ulster level before a Jackson penalty edged the home side ahead just before half time.
Toby Flood quickly levelled for Leicester, before Darren Cave tried to restore Ulster's lead, but had his try ruled out for a forward pass.
Another Jackson kick put the hosts back in front, while Flood missed a chance to level, slicing his penalty wide just three minutes later.
Jackson continued his solid performance with the boot, adding a further two kicks to open up a nine-point advantage with less than 10 minutes to play.
Owen Williams came off the bench in place of Flood late on, and managed to reduce the deficit with a late effort at the posts, but it was not enough to prevent the Tigers losing their opening game of the campaign.