Barca were struggling to break down a determined Bilbao side at the Nou Camp until substitute Neymar struck in the 79th minute, running on to a threaded Messi pass and scoring with a precise low shot into the corner.
Four-times World Player of the Year Messi set up the Brazil forward for his second five minutes later, dribbling down the right and squaring for an unmarked Neymar to pick his spot in the same corner.
"Messi is the best in the world, a star," Neymar, who came on for the final half an hour to replace teenage Spain forward Munir el Haddadi, told reporters.
"I am improving playing with him," added the 22-year-old, who started on the bench as he continues his comeback from an ankle injury.
Messi struck twice in Barca's 3-0 home win against Elche in their opening game of the campaign and set up Sandro Ramirez to score in their 1-0 success at Villarreal on their second outing.
Seeking to improve on last season's second-placed finish to Atletico Madrid, Barcelona have made a fine start and are the only team with a perfect record of three wins in three games.
Barca's new coach Luis Enrique echoed Neymar's praise of Argentina captain Messi and noted his prolific scoring record means the goals he creates for team mates are often eclipsed.
"Messi is the best not only because of his goals but also for his assists," Luis Enrique told a news conference.
"He does things in training that I have not even seen... on the Playstation," added the former Barca and Spain midfielder, who took over from Gerardo Martino at the end of last season. "It is a pleasure and a privilege to have him in the team."
Neymar said he had been feeling the effects of his trip to play for his country in friendlies against Colombia and Ecuador but was delighted with his performance against Bilbao.
After joining from Brazilian club Santos, Neymar blew hot and cold in his debut season in Spain but had an excellent World Cup before picking up a back injury that ruled him out of the semi-final defeat by Germany.
His connection with Messi did not always work as smoothly as hoped last term but they linked up well on Saturday and Neymar said he hoped this would be his breakthrough year.
"I am very happy with today's match, it went perfectly, with two goals," he said.
"We knew we were going to suffer a bit but we were able to perform well in defence so that later we could win the game.
"I returned from international duty feeling a bit tired, it was a long journey. I hope that this year is my year."
(Writing by Iain Rogers, editing by Ken Ferris)
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