Record 14-time Cup winners Arsenal fell behind in the third minute when Colby Bishop raised the decibel level at Fratton Park by firing in a rebound after a save by Kepa Arrizabalaga.
They were only behind for five minutes, though, as Arsenal levelled with their first of three goals from corners.
Portsmouth were unable to deal with the set-piece delivery and the ball eventually went in off home player Andre Dozzell.
Martinelli put the visitors in front when he glanced in Noni Madueke's corner in the 25th minute and then hit the post soon afterwards from close range.
Madueke put a penalty wide just before halftime to keep Portsmouth in contention, but Arsenal eased away after the break as Martinelli converted from a superb pass by Gabriel Jesus.
He completed his hat-trick with a header as Portsmouth were again unable to defend from an Arsenal set piece.
Arsenal have now scored 17 goals from corners this season.