The Shrimpers went in front in the 14th minute when Dougie Freedman back-heeled the ball to Barnard and he rifled into the bottom-right corner past the diving Mikkel Andersen.
The Pirates' best chance of the half came when Chris Dickson teed up Jo Kuffour and the striker beat two Blues defenders before seeing his 20-yard effort saved by Steve Mildenhall.
Barnard bagged his second from close range when he slipped home a cross from the right byline from Alan McCormack on 47 minutes.
Rovers stepped up the pressure but Dickson missed three good chances before Rovers pulled a goal back after 56 minutes when a sliding Anthony Grant put Wright's cross through his own net under pressure from Kuffour.
Dickson then fired straight at Mildenhall from close range, before Andersen parried a Franck Moussa shot. Rovers substitute Darryl Duffy was denied late on by Mildenhall and fellow sub Andy Williams saw his follow-up cleared.