Joao Neves opened the scoring after 16 minutes on Wednesday (Thursday AEDT) when he pounced on a rebound off the crossbar to score at the far post.
Ramos doubled the lead in the 36th minute with a close-range finish from Senny Mayulu's cross.
Ramos bagged his second from the penalty spot soon after the interval, before setting up Desire Doue to make it 4-0.
Doue then set up the Portuguese striker for his third goal, and the 19-year-old also teed up Mayulu for the sixth.
Substitute Ousmane Dembele struck a powerful shot to complete the rout five minutes from time, netting his first goal in the Cup and 25th in all competitions this season.
Meanwhile, Dunkerque showed their fighting spirit once again, rallying from 0-2 down to beat top-flight Brest 3-2 and reach the semis.
The second-tier side had already knocked out Ligue 1 high-flyers Lille in the previous round on penalty kicks after equalising six minutes into stoppage time in that game.
Dunkerque looked to be heading out against Brest, who played in the Champions League this season.
But they exploited Brest's weakness on free kicks and corners as central defender Vincent Sasso pulled a goal back midway through the second half, before defender Opa Sangante netted twice in the final 10 minutes.
Dunkerque and PSG join Reims and fourth-tier Cannes in the semi-final draw. Cannes had knocked out second-tier Guingamp on Tuesday in yet another upset.
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