Bearman's 80th minute try, after a quick turnover, confirmed the region's fifth consecutive win at Rodney Parade which will boost morale ahead of next week's Heineken Cup trip to Gloucester.
For Connacht, they will head into their European Challenge Cup opener at home to Madrid off the back of their fourth league defeat.
Having already made a raft of team changes, the Irish side had to tweak their starting line-up further before kick-off when centre Niva Ta'auso cried off with a hand injury and was indirectly replaced by Brian Tuohy.
Despite driving rain and a strong wind, the Dragons made a determined start to force a penalty from which James Arlidge kicked them into the lead.
The hosts did all the early attacking and the opposition back row of Mike McCarthy, Johnny O'Connor and skipper John Muldoon had to make a concerted effort to prevent further scores.
But they were at fault when allowing Danny Lee to break from a scrummage and feed Harris who scored the try and Arlidge banged over the conversion.
Conor O'Loughlin made a couple of strong runs for the visitors and some forward pressure gave them field position from where O'Connor raced over, only to be recalled by referee Peter Fitzgibbon for a previous infringement.
Some scrambling play on the Dragons' line led to the referee calling on the television match official for a touchdown decision, which went against the visitors.
But Connacht were not to be denied and Miah Nikora sliced through for a well-taken try under the posts that he converted himself.
Arlidge and Nikora exchanged penalties just before half-time to give Dragons an interval lead of 13-10 that was deserved on the balance of play.
With the wind at their backs for the second half, Connacht started to make use of the elements and Nikora was just wide with a drop goal attempt.
The slippery conditions made handling more difficult and scoring chances were missed on both sides.
Replacement Shaun Connor drove over a penalty to make it 16-10 and in a grandstand finish, Bearman raced 30 metres down the touchline to score with Connor converting.
|Newport Gwent Dragons Score Card|
|Connacht Rugby Score Card|