Jones and Aled Brew crossed for first half tries for the Dragons, who led 15-10 at the break. Back row duo Sam Warburton and Xavier Rush touched down for the Blues, drawing the play-off hopefuls level by the 46th minute.
But Coombs was driven over following a well-executed lineout and Jones' superb 50-metre penalty took the game beyond Cardiff's reach.
It was the Dragons' first win in nine meetings with the Blues, who were suffering their first away defeat since February. Most notably, it leaves the play-off race wide open heading into next week's final round of matches.
Munster and Leinster are already through to the semi-finals, with Ulster, Cardiff, the Ospreys and Scarlets fighting it out to join them.
Third-placed Ulster (63 points) are away to the Dragons in their final league game, while Cardiff, who are fourth on 60 points, are on the road to the Scarlets (57 points) and the fourth-placed Ospreys are Viadana-bound to face Aironi.
Having secured a bonus point against Benetton Treviso last week, the Blues wanted to wrap things up in Newport but the Dragons had other ideas and threw a spanner in the works right from the off.
It was also a red letter day for teenager Jones, who was drafted in from Cross Keys to make his first start for the men of Gwent at full-back.
The Welsh Premiership starlet, who is just 20, crowned his league debut with a seventh minute try. He burst through a gap in the Blues' defence between full-back Dan Fish and lock Bradley Davies to dive over in the corner.
Jones' smashing score reclaimed the lead for the Dragons after Cardiff out-half Dan Parks had landed the first points of the afternoon with a penalty.
By the quarter hour mark, the in-form Aled Brew created a slice of history, breaking the record for the most tries scored by any player since Welsh rugby went regional in 2003.
Brew's 16th try of the season came after a cross-field kick from number 10 Jason Tovey was knocked down by Luke Charteris into the winger's welcoming arms for a simple run-in from five metres out.
Given the close nature of the game and regional bragging rights on the line, it was no surprise that there was evidence of some niggle between the sides.
Cardiff number 8 Rush and Dragons flanker Gavin Thomas traded punches during a midfield brawl, and referee James Jones was quick to issue a warning to both captains.
Tovey kicked the Dragons further into the lead but, when home lock Adam Jones was sent to the sin-bin for preventing the ball from coming out of a ruck on his line, the Blues got back into the match.
Flanker Warburton immediately grabbed a try from a pick-up and drive by Rush off an attacking scrum.
Then, early in the second period, a lineout take five metres out by replacement lock Deiniol Jones saw the Blues pack drive over the line and Rush come up with the ball.
That levelled matters at 15-15 as the Blues took advantage of a lacklustre start to the half by the Dragons and David Young's side suddenly looked favourites to win.
But crucially they lost momentum and despite dominating the scrums, they could not get over the line again.
The Dragons retook the lead when a lineout within metres of the visitors' whitewash saw Charteris take the throw and he helped his pack drive Coombs over for a converted try.
That made it 22-15 and Jones' long range kick gave the Dragons a 10-point gap, moving them within touching distance of their third win in four league games.
Lively scrum half Evans put the icing on the cake for the home side with a drop goal from close range, right at the death.
|Newport Gwent Dragons Score Card|
|Cardiff Blues Score Card|