Arsenal striker Gabriel Jesus, who arrived from Manchester City in the close season, looked set to get off to a flying start by dancing through the Palace defence only to see his shot blocked, with Martinelli dragging the rebound wide of the post.
Oleksandr Zinchenko, another new signing from City, also looked lively and had an early shot blocked as Arsenal quickly raised the tempo but the Ukrainian soon had a hand in the opening goal as Palace were caught flat-footed at a corner.
Bukayo Saka's inswinger found Ukraine international Zinchenko in acres of space and he headed the ball back across the goal where Martinelli nodded it into the net.
Palace should have levelled before the break as Odsonne Edouard headed goalwards, but visiting goalkeeper Aaron Ramsdale pulled off a superb reflex save to preserve his side's lead.