The win against 11th-placed Wolves, extends Burnley's unbeaten run to four games and moves them up to 16th spot on 13 points and one behind Arsenal with a game in hand.
Burnley had been struggling for goals, managing just six in their opening 12 league matches of the season, and manager Sean Dyche turned to his trusted old pairing of Barnes and Wood for the visit of Nuno Espirito Santo's team.
Espirito Santo could face sanctions after he took aim at referee Lee Mason despite the late penalty and his words are likely to be reviewed by the league's disciplinary body.
"The referee did not have the quality for a Premier League game, this is a problem we know we had Lee Mason before. It is not about the crucial mistake or decisions, it is about how he handles the game," said Espirito Santo.
"The players get nervous, he whistles from the noises from the players. We are talking about the best competition but he does not have the quality to whistle the game.
"I just don't want to see him more, I told him I hope he does not whistle a game of ours again. He cannot control the players, the players are constantly arguing. The other referees let the game flow," he said.
Barnes, who last scored in November 2019, put the Clarets ahead 10 minutes before the break, heading in at the back post after Charlie Taylor burst to the byline and floated in a high cross.
Josh Brownhill went close for the home side after the break with a well-struck effort from outside the box which crashed against the bar.
But moments later Burnley had their second when Ben Mee rose to head a free-kick across the face of goal and, with Barnes causing havoc in the Wolves defence, the ball fell to New Zealander Wood who fired home.
Burnley looked more determined with Westwood and Brownhill relentless in midfield but, having failed to take their chances to kill off the game, were forced to sweat in the final minutes after an 89th minute Fabio Silva penalty gave Wolves hope.
Burnley brought on young midfielder Josh Benson and, before the subsitute had even touched the ball, he brought down Silva in the box and the 18-year-old club-record signing converted.
But the Clarets, who now have just one defeat in their last seven, held on for three vital points in what looks set to be a tight battle at the bottom end of the table.
Barnes, who has had injury problems during his barren spell, justified his recall with a typically active and aggressive performance and was delighted to have got on the scoresheet.
"It feels brilliant to hit the back of the net. It has been a difficult period for myself. Frustration plays a part but I know my ability so I had to keep going and [thought] the goal would come," said the striker.
Dyche was surprised Wolves' Portuguese midfielder Joao Moutinho was not sent off after a VAR review of a challenge on Josh Brownhill in the final minutes but was more supportive of the officials.
"I've no problems with the referees, it is really difficult now. The three most difficult jobs are the two managers and the referee. I don't think any game is easy for referees at the moment. We all want the decisions of course, it is one of those things," he said.