Argentina's perfect record was ended in a draw against Paraguay last time out, however, the Albiceleste stayed undefeated on the road to Qatar 2022 thanks to Gonzalez and Martinez in Lima on Tuesday.
Gonzalez, who scored in the Paraguay stalemate, opened the scoring with his second international goal in five appearances in the 17th minute before Martinez doubled Argentina's lead 11 minutes later.
With the result, Argentina remain second and two points behind CONMEBOL leaders Brazil through four matchdays.
Peru started on the front foot at the Estadio Nacional, where Christian Cueva appealed for a penalty inside the ninth minute but it was waived away by the referee, despite a pitch-side review – Argentina goalkeeper Franco Armani was adjudged not to have made contact.
Pedro Gallese did not have much to do, but he was forced to collect the ball from his net after Gonzalez capped a fine team goal – Giovani Lo Celso creating space in a sublime movement before laying off a pass with the outside of his boot as the former finished powerfully across the Peru goalkeeper.
Lionel Messi almost helped double Argentina's lead when he played a throughball for Nicolas Tagliafico, who was denied by a last-ditch block in the 24th minute.
Argentina, though, did get their second goal four minutes later after Martinez broke the offside trap, running onto Leandro Paredes' pass down the middle and rounding Gallese in a swift attacking move.
While Peru ended the half strongly, they had nothing to show for it, though they continued to build some momentum in the second half.
Lucas Ocampos almost scored a goal with his first touch, but Gallese made a fine save approaching the hour-mark, after Messi's exquisite pass almost released Martinez.
Messi tried desperately to get on the scoresheet, but he had his penalty appeals waived away and his goal-bound effort was deflected for a corner, while substitute Angel Di Maria was thwarted by Gallese.
What does it mean? Argentina roll on
Argentina are yet to taste defeat in four World Cup 2022 qualifiers. Not since last year's Copa America in July have Lionel Scaloni's side lost – a run of 11 international fixtures. It was also Argentina's first victory away to Peru since September 2004, having drawn three times previously.
Lo Celso shines in Lima
After impressing off the bench against Paraguay, Lo Celso was at it again from the outset on Tuesday. He helped set the tone with a wonderful assist – his quick feet, vision, composure and passing on display.
Subdued Messi wasteful
Messi was far from his best against Peru, but it did not matter as the likes of Lo Celso and Paredes stepped up. The six-time Ballon d'Or winner was not without his chances, though Argentina's all-time leading goalscorer uncharacteristically saw two golden opportunities go begging in the second half. Messi has now gone three international matches without scoring.
Argentina will be back in action in March next year, the South American powerhouse facing rivals Uruguay and Brazil, while Peru will take on Bolivia and Venezuela.