The visitors made a nervous start against the wind and rain at Rodney Parade, where the Dragons had won their last four matches.
Fly-half James Arlidge put the Dragons ahead with a third-minute penalty.
In-form winger Richard Fussell sparked an exciting attack for the home side that passed through numerous sets of hands, ending when prop Gethin Robinson was held up over the line.
James Harris scores the opening try of the night
Bearman exploded off the resultant five-metre scrum before confounding the defence with his delightful pass that gifted flanker James Harris a simple stroll home.
The Dragons struggled to contain a chip through from the first major Irish attack, the video referee called in to play before ruling out Conor O'Loughlin's grounding because of a Connacht knock-on.
Nikora soon made amends, though, attacking from solid scrum ball in the Dragons' 22.
The New Zealander stepped inside a weak challenge from opposite number Arlidge and slid in, converting his own score to reduce the gap to 10-7.
Arlidge's second penalty was cancelled out by a Nikora kick, the final action of the first half.
The Dragons replaced their fly-half with Shaun Connor early in the second stanza, and were forced into another change when tight-head Robinson went off with a shoulder injury, increasing their concerns at prop ahead of Friday's daunting Heineken Cup trip to Gloucester.
Connor extended the home side's advantage to 16-10 with a 65th-minute penalty, but the Irish stayed competitive as they sought a winning score.
To play as poorly as that and win shows that we have come on strides
Dragons coach Paul Turner
But their ambition cost them a hard-earned bonus point as they attacked from deep in their own half.
The Dragons won the ball on the floor and quickly fed Bearman.
The English-born back-rower - who completed his Wales residential qualification this year - immediately hared for the corner, surely impressing watching Wales coach Warren Gatland.
Dragons coach Paul Turner: "To play as poorly as that and win shows that we have come on strides.
"Last year Connacht would have pinched a game like that, it was a good win for us."
Dragons: Hugh Gustafson, Steve Jones, Gethin Robinson, Adam Jones, Luke Charteris (capt), Dan Lydiate, James Harris, Joe Bearman, Danny Lee, James Arlidge, Richard Fussell, Tom Riley, Rhodri Gomer-Davies, Aled Brew, Martyn Thomas. Replacements: Hoani MacDonald for Dan Lydiate (40), Dan Lydiate for Hoani MacDonald (42), Nigel Hall for Gethin Robinson (45), Hoani MacDonald for James Harris (50), Shaun Connor for James Arlidge (51), Wayne Evans for Danny Lee (60).
Connacht: Ronan Loughney, Sean Cronin, Jamie Hagan, Mike Swift, Bernie Upton, Mike McCarthy, Jonny O'Connor, John Muldoon (capt), Connor O'Loughlin, Miah Nikora, Liam Bibo, Keith Matthews, Troy Nathan, Brian Tuohy, Gavin Duffy. Replacements: Andrew Browne for Mike Swift (50), Brett Wilkinson for Ronan Loughney (53), Frank Murphy for Connor O'Loughlin (60), Mike Swift for Andrew Browne (65), Ian Keatley for Miah Nikora (70), Andrew Browne for Mike Swift (78), Andrew Browne for Mike Swift (78).
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