Defender Javier Zanetti won an Argentine record 140th cap in the match, which his team controlled at will against mediocre opponents.
Striker Ezequiel Lavezzi put the Copa hosts ahead in the fifth minute, sticking the ball over goalkeeper Arjan Beqaj from Messi's perfect pass, before Messi made it 2-0 in the 43rd, taking the ball from Lavezzi's lay off and striking a low left-footed shot inside the base of Beqaj's post.
"Lionel had a great match, generating scoring chances and scoring," coach Sergio Batista said of Messi, who took his international tally to 17 in 57 matches.
"The team played well. There are things that need adjusting because our opponents did not attack much and did not see much of the ball, but I'm happy because we showed everything we've been practising," he told reporters.
SUBSTITUTE SCORERS
Argentina got their third in the 74th minute after Messi fed another sublime pass with the outside of his left foot to Sergio Aguero, who rounded the goalkeeper to slot home from a tight angle.
"These kind of matches help us to get confidence and get to know each other better," said Lavezzi.
Aguero then laid on the last goal in the final minute for fellow substitute Carlos Tevez, who was recalled by Batista despite appearing out of the Copa reckoning less than a month ago.
Messi drew three defenders as he raced into the box, leaving Aguero to pass to Tevez to score with a low shot.
Albania did not create a single clear-cut chance, goalkeeper Sergio Romero going down just once to stop an early effort.
"It was not as hard as it's going to be in the Copa [America] but it's useful to get our movements right," Tevez told Argentine TV.
The Copa America, which Argentina have not won since 1993, kicks off on July 1 when the hosts play Bolivia in La Plata in their Group A opener. The final is scheduled for the Monumental on July 24.
