Neves, the youngest player to start a league match for Porto at the age of 17 years, five months and two days, opened the scoring after just 12 minutes at Estadio do Dragao on Friday.
His strike looked set to be the only goal of the game, but Jackson Martinez added a second for the 27-time Portuguese champions in the fourth minute of injury time, rounding off an encouraging first competitive game for new coach Julen Lopetegui.
Spaniard Lopetegui is tasked with reviving Porto after the club finished 13 points behind champions Benfica last term.
And the 47-year-old could hardly have wished for a better start, with Neves keeping his cool to drill home a first-time shot after an early corner had only been half-cleared by the Maritimo defence.
Fellow Porto debutants Bruno Martins Indi and Yacine Brahimi had opportunities to extend Porto's lead prior to the interval, while Maritimo - sixth in the Primeira Liga last term - saw chances of their own go begging.
The second half brought less goalmouth action, but the hosts did eventually double their lead.
Cristian Tello, another new face at Porto following his loan move from Barcelona, was the creator, jinking in from the right before playing a delicate reverse pass into the path of Martinez, who converted at the second attempt after his initial strike was saved by Romain Salin.
Victory ensures Porto will head into the first leg of their UEFA Champions League play-off tie against Lille, which takes place at Stade Pierre-Mauroy on Wednesday, in good spirits.