New Southampton striker Che Adams went close early, sliding in at the back post to meet a dangerous cross from Yan Valery but the £15 million signing from Birmingham City couldn't find the target.
Burnley, who left their own close season signing Jay Rodriguez on the bench against his former club, had the ball in the net through Chris Wood in the 16th minute but provider Barnes was flagged offside and a Video Assistant Referee review backed the decision.
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It was a scrappy affair in wild winds and driving rain but the breakthrough came in the 63rd minute when Saints defender Jannik Vestergaard misjudged a hopeful long ball forward from Erik Pieters and Barnes latched on to it and fired home.
The pair combined again for the Clarets' second, seven minutes later with Dutch debutant Pieters' cross from deep met superbly on the volley by Barnes.

Ashley Barnes celebrates with fans

Image credit: Reuters

Icelandic winger Johann Berg Gudmundsson made it 3-0 in the 75th, winning a challenge with Ryan Bertrand and bursting goalward before beating goalkeeper Angus Gunn with a precise finish.
"We lost it completely," said Saints manager Ralph Hasenhuttl, who suffered a blow before the game with defender Pierre-Emile Hojbjerg restricted to the bench after suffering a pre-match illness.
"We lost it for 12 minutes and now this game is gone. Three-nil down, no chance to come back. We knew it would be difficult here. We knew it is not easy here. We did it ok for 60 minutes and then it is not good enough," added the Austrian.
"Really good start, really pleasing start," said Burnley manager Dyche.
"The players have got their reward for a really pleasing pre-season. There has been a strong desire and super fitness levels. I think there is an intent about the group," he said, adding particular praise for the 29-year-old Barnes.
"His finishes were excellent. He continues to learn about himself and the game. He is certainly a very effective player," Dyche said.
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