Striker Gonzalo Higuain scored an 11th-minute winner for the title favorites, which had already secured a spot in the quarterfinals of the South American tournament.
Argentina reached seven points with the victory in Group B, two points more than Paraguay, which earlier in the day drew Uruguay 1-1 in La Serena. Defending champion Uruguay ended with four points but advanced as one of the best-two third-place finishers.
Jamaica, eliminated before the match began, finished the tournament with zero points and no goals scored.
By finishing first in the group, Argentina will get to play the quarterfinals against the second-best third-place finisher in all three groups.
Argentina is trying to end a 22-year title drought. It hasn't won a significant trophy since the 1993 Copa America in Ecuador, when Messi was only six years old.
"We reached the first goal, which was to advance in first place," Messi said. "We played a great first half and we could've scored more, but in the second we didn't have the same intensity, everything was slower. In the knockout matches we can't let that happen."
Messi put in a lackluster performance in his milestone match. His most significant chance came in the 56th, when his lob shot from the top of the area was barely stopped by the outstretched arms of Jamaica goalkeeper Dwayne Miller.
The mediocre performance didn't affect Messi's popularity, though. After the final whistle, Jamaican striker Deshorn Brown was seen taking a selfie with the Barcelona star before the teams left the field.
Argentina outplayed Jamaica from the beginning at the Sausalito Stadium. It created a series of scoring chances, especially in the first half, but had difficulties capitalizing on the opportunities. Twice it hit the crossbar of goalkeeper Miller.
"The second half was boring," Argentina coach Gerardo Martino said. "We were in control but didn't have enough quickness to be able to threaten Jamaica or any other rival. If we play like this in future games we will be in trouble."
Higuain, in his 50th match with the national team, scored the winner with a right-footed strike from near the penalty spot, turning in front of three Jamaican defenders before sending a low shot into the net after a pass by Angel Di Maria.
Jamaica, focused on the upcoming Gold Cup, tried to push forward near the end but it wasn't enough to get the equalizer.
"In the first half we had too much respect for Argentina, the second-best team in the world," Jamaica's German coach Winfried Schafer said. "The players were nervous facing Messi, Di Maria, (Javier) Mascherano."
The Argentines won the previous four Copa Americas played in Chile, the last in 1991. The World Cup finalists hosted the South American tournament in 2011, but Messi and his teammates were eliminated by eventual champion Uruguay in the quarterfinals. Argentina lost consecutive Copa America finals to Brazil in 2004 and 2007.
Argentina wins record 16th Copa America title, beats Colombia 1...
Paraguay, Uruguay draw 1-1 and advance to Copa quarters
LA SERENA, Chile: " Defending champion Uruguay and Paraguay drew 1-1 in their last group game and both advanced to the Copa America quarterfinals on Saturday.
Jose Maria Gimenez put Uruguay in front in the 29th minute on a header, and Lucas Barrios leveled with his own header a minute before halftime.
Both teams had chances in the second half, though neither grabbed control. The game was a rematch of the 2011 final in Buenos Aires, where Uruguay won 3-0.
The draw suited both teams, though defender Gimenez said that was never the intention.
"There was a lot to play for " the prestige, wearing the national team shirt," Gimenez said. "None of the teams here are playing for a draw, nor to lose."
Edinson Cavani, Uruguay's leader with Luis Suarez banned from the tournament, has done little so far. He had at least three clear openings to score. On his best chance, Cavani was stopped in the 68th by Paraguay goalkeeper Justo Villar on a point-blank shot.
Paraguay has had a strong tournament, drawing with Uruguay and Argentina.
"I'm very happy to show off the strength that Paraguay has," Argentina-born Barrios said. "It's an important team, and we've shown that right from the first match. We are stronger than ever."
In Group B, Paraguay has five points, and Uruguay four. Argentina, on four, also advanced before its late game against Jamaica.
Bolivia and Chile advanced on Friday from Group A, with Group C matches wrapping up on Sunday.
The quarterfinals begin on Wednesday.
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