Juan Roman Riquelme, Argentina midfielder, scored twice and set up another as the Albiceleste produced a display of attacking football that brought back memories of their 6-0 win over Serbia & Montenegro at last year's World Cup.
Mexico were more opportunist, as they dominated their match from the fifth minute when Paraguay goalkeeper Aldo Bobadilla gave away a penalty and was sent off before Nery Castillo made sure of the spot kick.
Alfio Basile, Argentina coach, summed up his team's philosophy after they scored four times in the second half.
"It's not just about winning. You can simply win a game, but to win the way we are doing makes me proud," Basile said.
"We played patiently, moving our rivals around all over the pitch."
Riquelme's first came just after halftime when he collected Carlos Tevez's pass and fired a left-foot shot low into the corner, before splitting the Peru defence with a deft pass to Messi, who rounded Leao Butron to score his first of the tournament in the 61st minute.
Argentina added two more in the last 15 minutes through Javier Mascherano and Riquelme, who finished off a counter-attack in delightfully nonchalant style for his fourth
goal of the competition.
"We gave it our best effort but we were well and truly beaten. It's part of football," Julio Cesar Uribe, Peru coach, said.
"Argentina had the variations and superior talent."
Mexico take full advantage
Meanwhile at the Estadio Monumental de Maturín, Mexico took full advantage of Bobadilla's dismissal, with replacement keeper Joel Zayas' first job on his international debut being to pick Castillo's penalty out of the net.
Castillo added another goal before halftime, his fourth of the tournament, while midfielder Gerardo Torrado was on target in between.
Paraguay kept the score relatively respectable until the last 15 minutes when Fernando Arce, Cuauhtemoc Blanco, from another penalty, and Omar Bravo doubled Mexico's tally.
"The target is to win the Copa America and we're on the right path, especially considering that we're improving with every game," Hugo Sanchez, Mexico coach, said.
Mexico have been taking part in the Copa America as guests since 1993 and have already reached two finals, losing to Argentina on their debut and Colombia in 2001.
Mexico will face Argentina in their semi-final on Wednesday in a repeat of last year's second round match at the World Cup, while Uruguay meet Brazil on Tuesday.