Messi created chances for Alexis Sanchez to score in the sixth minute and Cesc Fabregas in the 55th for a 2-0 win away to Benfica.
Barcelona's comfortable night was tarnished late on as the four-time champion lost captain Carles Puyol who dislocated his left elbow after landing awkwardly. Midfielder Sergio Busquets was then shown a straight red card for lunging into challenges too aggressively.
Van Persie scored with a fortunate finish in the 29th and a classy shot in the 49th, helping Man United come from behind to beat CFR Cluj 2-1 in Romania.
"Wayne (Rooney) gave me the good pass, so I have to thank him," Van Persie said of his teammate's vision to create the winning goal. "Not many players can give a pass like that."
Defending champion Chelsea beat Nordsjaelland 4-0 in Denmark, and shares Group E leadership on with Shakhtar Donetsk which drew 1-1 away to Juventus.
Bayern Munich, the beaten finalist last season, lost 3-1 to BATE Borisov in Belarus, Valencia beat Lille 2-0, and Braga won 2-0 at Galatasaray.
In the early kickoff in Barcelona's group, Celtic scored a 90th-minute goal to win 3-2 at Spartak Moscow which led in the second half before having defender Juan Manuel Insaurralde sent off.
Messi has started the season in sparkling form in Spain, and the Argentina forward was a creative force at the Stadium of Light in Lisbon.
The Argentina forward hit a fierce low cross from the left that Sanchez poked in from close range, and was denied by a smart low save by Benfica goalkeeper Artur in the 22nd.
Messi helped ensure victory when, after luring a group of defenders toward him, he laid off a pass to Fabregas in space to fire a left-foot shot into the net.
Puyol was carried off on a stretcher when he falling heavily on his left arm after leaping to meet a corner in the Benfica penalty area in the 75th.
Barcelona leads Group G with six points, two clear of Celtic which it plays next in Spain in three weeks' time.
Celtic led in the first half through Gary Hooper, but fell behind to two goals from Spartak's Nigerian forward Emmanuel Emenike. With a player advantage, the Scottish champion levelled with an own goal by Dmitry Kombarov then won through a Georgios Samaras header.
"They were magnificent and they played magnificent football at a very tough venue against an excellent side," Celtic manager Neil Lennon said.
Man United trailed to a 14th-minute shot from Pantelis Kapetanos for Cluj, but turned around the match when Van Persie converted crosses by Wayne Rooney.
The Dutchman was perhaps lucky when an attempted header looped off his right shoulder and into the net, though his second goal was a deft first-time volley from Rooney's chipped pass.
Man United has a maximum six points, three ahead of Cluj and Braga, which scored goals through Ruben Micael and Osorio Alan in Turkey.
Chelsea was flattered by the margin of its victory after the Danish champions began the game brightly. Juan Mata scored in the 31st and David Luiz lashed in a free kick in the 79th to ease nerves. Mata and Ramires added late goals.
"Until the 75th minute, we played tremendously. After that we stopped," Nordsjaelland coach Kasper Hjulmand said.
Chelsea, which drew at home to Juventus last month, has four points along with Shakhtar, which led in Turin through Alex Teixeira in the 23rd. Two minutes later, Leonardo Bonucci levelled.
BATE stunned Bayern to take control of Group F with its second straight 3-1 win, after beating Lille last month.
The Belarus champion scored through Aleksandr Pavlov in the 23rd minute and Vitali Rodionov netted the second in the 78th. Bayern got a stoppage-time goal from Franck Ribery but Renan Bressan then sealed the win.
"We always believed that we could win this match," BATE forward Aleksandr Hleb said. "We were confident even after we conceded the late goal."
Valencia's Jonas scored in each half Jonas against Lille, which had defender Mathieu Debuchy sent off.Suggest a correction