Barnes cancelled out Matt Ritchie’s first-half strike after lashing a half-volley past Cherries goalkeeper Lee Camp.
Barnes’ 55th-minute shot saw him bring the ball under control from a throw-in with his back to goal before angling a left-footed effort into the net.
It was his fourth goal of the season and his equaliser ensured Bournemouth secured their first draw at home this season.
Ritchie marked his first start of the campaign by putting Bournemouth ahead with a thunderous 20-yard strike from a free-kick by Charlie Daniels in the 29th minute.
Ritchie was sent to Italy by Bournemouth manager Eddie Howe in a bid to cure a thigh muscle injury which had kept him out for four months.
The former Swindon winger responded with his stunning goal and almost added another in a first half dominated by the home side.
Ritchie was almost on the scoresheet three minutes later when Ryan Fraser, also restored to Howe’s line-up, centred from the left and Ritchie’s shot at full stretch was scrambled to safety by goalkeeper Tomasz Kuszczak.
Kuszczak was fortunate in the 41st minute when a deflected free-kick from Ritchie went narrowly past his post, with the former Manchester United stopper beaten.
Brighton, rattled by the setback, had chances before Ritchie’s goal with Gordon Greer slicing his effort over from close range after being found by Inigo Calderon.
Scottish international Craig Conway set up an opportunity for Barnes in the ninth minute but the Brighton striker failed to connect.
Brighton boss Oscar Garcia introduced Liam Bridcutt for Jake Forster-Caskey at the break and the move quickly paid off with Barnes’ goal 10 minutes later.
Bournemouth refused to wilt at Brighton’s ascendency and had chances of their own to restore their lead.
Ritchie attempted to turn provider with a defence-splitting pass which found Brett Pitman after 50 minutes, but Kuszczak darted from his line to make a crucial block.
Former Brighton defender Steve Cook headed across goal and narrowly wide from a floated Pitman free-kick.
Pitman was involved again in the 74th minute when he raced on to a Eunan O’Kane pass but dragged his early strike across the face of goal.
Busy Ritchie almost made a dream return to the starting line-up with an injury-time winner but his 25-yard shot fell just past the post.
The nearest the visitors came to a second was in the 87th minute as on-loan Conway saw his shot gathered by the diving Camp.
Brighton were still able to extend their unbeaten run to six matches, while Bournemouth will regret not making the most of their first-half dominance.