Nicolas Gonzalez put the visitors ahead with an angled shot in 17 minutes following incisive work from Giovani Lo Ceslo on the left flank and then Lautaro Martinez doubled their lead 11 minutes later when he ran on to a superb pass from Leandro Paredes and rounded the keeper to slot home.
"From the very start we played a great game, the goals came and we created lots of chances," said captain Lionel Messi.
"Happy with the victory, we needed it after the game we played the other day," he added in reference to their 1-1 draw at home to Paraguay last week.
"This is the path we have to keep on going down."
Argentina pressured after the break with goalkeeper Pedro Gallese denying Lucas Ocampos with a great block and Anderson Santamaria intervening in the box to stop Messi from adding a third on the counterattack.
The win extended Argentina's unbeaten run to 11 games and means they stay second in the 10-team South American qualifying group for Qatar 2022, two points behind leaders Brazil.
It also marked the first time since the year 2000 that the two-times world champions have won three consecutive away qualifiers.
Peru's defeat extended their winless run to seven matches under coach Ricardo Gareca, the Argentine who guided them to their first World Cup finals in 36 years last time around in Russia.
Argentina's next two qualifiers are in March, at home to Uruguay and away to Brazil, the only side in the group to have won all four of their matches so far.
The top four teams in the 10-team South American group qualify automatically for Qatar 2022 and the fifth-placed side goes into an inter-regional playoff.
Earlier in the night, Bolivia were denied their first win when Kaku scored with 18 minutes remaining to rescue a 2-2 draw for home side Paraguay in Asuncion.
Bolivia last qualified for the World Cup in 1994 and were 2-1 up at half time but they had to settle for a point that still leaves them bottom of the group.
