Jackson, discarded by Joe Schmidt for Sunday's Six Nations opener with Wales, came off the bench at half-time and held his nerve to put Ulster in front for the first time in the match with just three minutes remaining.
The win takes Ulster a point above Scarlets and into first place, while the Dragons - who were down to 13 men at one point in the second half - remain 10th after at least picking up a losing bonus point.
The Dragons opened with a fourth minute penalty from Angus O'Brien after Ricky Lutton was penalised at a scrum under Ulster's posts.
They then burst out of their own 22 after Ulster spilled the ball, with wing Ashton Hewitt breaking several tackles before Carl Meyer kicked through for Adam Hughes to score in the left corner after 16 minutes.
O'Brien landed a superb conversion and the Dragons had a 10-0 lead.
The home side needed to respond but Sean Reidy was held up over the line and then Stuart Olding had a foot in touch when his inside pass sent fly-half Ian Humphreys through.
They did get over the line five minutes before the half-hour mark when Reidy broke Hughes' tackle to cross in the left corner for his third try in as many games. Humphreys then sank the conversion and Ulster had cut the Welsh side's lead to 10-7.
Humphreys then crossed from an intercept only for referee Nigel Owens to call it back for an earlier infringement. The half ended with O'Brien missing a drop goal and the Dragons having to be content with their three-point lead.
Meyer found a way over the line six minutes after the interval, finishing off a break from Nic Cudd, but crucially this time O'Brien missed the conversion.
Ulster responded when Ruan Pienaar was high-tackled and the home side - now with Nick Williams on as well as Jackson - went for the corner.
Owens awarded a penalty try after the Dragons collapsed the subsequent rolling maul and Matthew Screech was also sent to the sin bin.
Jackson converted the 52nd minute score to reduce the deficit to a single point.
Ben White then joined Screech on the sidelines after swinging an arm on Peter Browne.
Surprisingly it took until the 63rd minute before the home side were able to take advantage, but Roger Wilson's score was ruled out by the TMO for an earlier knock-on which occurred just before Screech returned to bring the Dragons back to 14 men.
Newport then survived without conceding and Cory Hill's arrival off the bench brought them back to full strength.
With eight minutes remaining, Jackson squandered a penalty opportunity by hitting the bar and watching it bounce out but he made no such mistake minutes later.