The 31-year-old scored twice and set up two more as Julen Lopetegui's men enjoyed their return to Estadio Do Dragao, with Oliver Torres, Vincent Aboubakar and Real Madrid new boy Danilo joining Quaresma on the scoresheet.
Maritimo had come from behind to defeat Porto 2-1 in the Portuguese league cup on Thursday, making it two games without a win in all competitions for Lopetegui's side, but they bounced back emphatically at home.
Having drawn at Nacional in the Primeira Liga before the international break and then losing away to Maritimo in the Taca da Liga, Porto looked more comfortable in front of their home fans.
Torres opened the scoring in the 33rd minute, stooping at the back post to head Quaresma's cross into the net from a tight angle.
Quaresma's delivery from the right wing saw the hosts double their lead in first-half stoppage time, as Aboubakar touched the ball home from up against the back post.
It was 3-0 seven minutes into the second half, as Quaresma took advantage of what looked like a fortunate penalty awarded to Yacine Brahimi to score from the spot.
There was some doubt about the validity of Danilo's goal in the 70th minute too, as the Brazilian full-back flirted with offside when receiving a return pass from Aboubakar before finishing calmly past Estoril goalkeeper Vagner.
Quaresma capped off Porto's victory six minutes later, winning the ball from Mattheus on the edge of the area, before producing a step-over to leave Vagner in his wake and finish into an empty net.
Porto's win kept the pressure on Benfica, as they stay within three points of the Primeira Liga leaders.
In Monday's other match, Vitoria Setubal came from behind to draw 1-1 at Maritimo despite having Adama Sene sent off.
In a dramatic finish at the Estadio dos Barreiros, Patrick Bauer put Maritimo in front in the 80th minute, before Sene was shown his second yellow card three minutes later.
But Zequinha earned Setubal a point with one minute remaining.
