Coutinho opened the scoring in the 13th minute from the penalty spot to halve the the aggregate deficit.
Fifteen minutes later, Ever Banega had the chance to restore Sevilla's two goal lead. But the Barcelona goalkeeper Jasper Cillessen saved his penalty.
The hosts made Banega pay for his lack of precision a few minutes later when Ivan Rakitic poked in a second.
After the pause, Coutinho headed in his second of the match to make it 3-0 and Sergi Roberto added a fourth.
Guilherme Arana 's strike mid way through the second-half gave Sevilla hope at 4-1. But Luis Suarez and Lionel Messi struck late to seal the comeback.
"It was said that we had given up in the cup, that we didn't want it," Messi said after the rout. "But in no way is this the case. This team wants to fight for all three titles, as is the obligation of Barca every year."
Barcelona join Valencia and Real Betis in the last four. Real Madrid are favourites to join them. Santiago Solari's men carry a 4-2 lead into their second leg at Girona on Thursday night.