Despite a 41-7 victory in the teams' most recent European meeting two years ago, the opening home tie against Leicester was always a potential banana skin for Mark Anscombe's men.
However, Ulster kept their balance impressively thanks to concentration, discipline and no little invention from numbers 1 to 23.
Ulster got into gear early, working the game's first real chance in the third minute which only broke down when Tommy Bowe slightly overthrew his final pass to Darren Cave - stationed in the Monaghan man's habitual position on the wing - and the ball looped just beyond Cave's grasp and out of play.
Leicester struck back with interest four minutes later, prop Logovi'i Mulipola picking up Dan Bowden's long pass into the corner and resisting Luke Marshall's desperate lunge to flop down by the flag.
Toby Flood converted the try, but Ulster recovered quickly from the early setback. Jared Payne's lengthy kick to touch put the Tigers under pressure on their own lineout five metres from their try-line.
The Aviva Premiership outfit cleared their lines cleanly in the first instance, but a subsequent Ulster lineout from a similar distance on the other side of the pitch paid dividends.
Bowe collected Jackson's searching kick to the left corner to ground a typically acrobatic try, with the in-form out-half adding the extras from the acutest of angles.
Tigers captain Flood knocked over a short range penalty to edge his side back in front on the 20-minute mark. Yet, those three points were cancelled out 10 minutes later by Jackson after some powerful ball-carrying into the 22 from a fired-up Rory Best.
A rare reversal at the Leicester lineout on 34 minutes failed to materialise into points when the Ulster forwards - with flanker Roger Wilson producing a man-of-the-match display - knocked on as they tried to barge their way through.
Nonetheless, som sublime work from Paul Marshall at scrum time, which saw the diminutive scrum half leave four or five Tigers in his wake on a mazy run, provoked the infringement which allowed Jackson to send his side in at half-time with a slender 13-10 lead.
The second half opened with two disallowed tries for Ulster, as the hosts put together a spell of cohesive and creative rugby hitherto unseen in the encounter.
The first saw an heroic length of the pitch dash from Jackson after he had picked up an errant Tigers pass deep in his own 22 - only for the out-half to find referee Romain Poite had called play to a halt for an apparent offside as he had seized possession.
The decision proved decidedly unpopular with the majority of the capacity Ravenhill crowd, all the more so as Flood duly dispatched the resulting penalty to bring his team level at 13 points apiece.
Ruan Pienaar, who recently committed to three more seasons with the Ulstermen, entered the fray in place of Marshall on 48 minutes, but it was Cave who hit the headlines in the very next move as he broke the Leicester line to ground beneath of the posts - only for the TMO to confirm that Jackson's pass to Payne in the build-up had gone forward.
Compensation of sorts could have come on 54 minutes with a further penalty attempt from Jackson, but the kick failed to trouble the posts - representing the number 10's first miss in nine consecutive attempts.
With Stuart Olding and Iain Henderson brought into the mix on the hour, Ulster’s verve showed no signs of abating.
And with Leicester number 8 Jordan Crane sent to the sin-bin for an intentional trip on Pienaar as he attempted to play the ball at the breakdown deep in the visitors' 22, Jackson made no mistake with his close-in penalty.
Flood then screwed an elementary penalty of his own wide on 63 minutes, and with Richard Cockerill's charges penalised again in their scrum moments later, Jackson put a little more clear blue water between the teams from the left wing with a low but accurate effort.
Brave play from Henderson kept the next Ulster move going as he dived head first on a loose ball and a flailing Leicester boot, and his courage was rewarded as Leicester again infringed in front of their posts, handing Jackson penalty number five on a plate.
More excitement came on 74 minutes as Owen Williams' drop goal attempt rebounded off the upright back into open play, although the replacement out-half reduced the gap to six points courtesy of a well-struck penalty two minutes later.
The final Ulster manoeuvre of the match then came to a premature end as Andrew Trimble uncharacteristically knocked on a simple ball in midfield, but with the clock at 80 minutes and a bonus point in their grasp Leicester simply kicked into touch after Ulster infringed at the subsequent scrum.
Next up for the Ulstermen is a trip to southern France to face in-form Top 14 outfit Montpellier, where a performance of at least equal calibre will be required to maintain the momentum established in round 1.