After Ivan Rakitic headed Sevilla in front five minutes in, Medel doubled the host's advantage by side-footing home Jesus Navas's pass in the 27th.
Deportivo pulled one back in the 32nd when Sevilla midfielder Geoffrey Kondogbia scored an own goal while trying to intercept a pass for Ivan "Riki" Sanchez in front of the goalmouth.
But Medel sealed Sevilla's fourth straight victory at Sanchez Pizjuan stadium when he volleyed home a cross from Alberto Moreno five minutes before halftime.
Deportivo created various chances in the second half, only for goalkeeper Antonio "Beto" Bastos to come up with two big reflex saves down the stretch to protect the three points.
"Sevilla was simply better than us," Vazquez said. "My team never gave up, but even so, when you play against a team that is better this is what happens."
Deportivo, which spent last season in the second division, remained winless on the road this season and six points from safety after recording its fifth consecutive loss. The struggling club can fall farther into trouble over the weekend after the rest of the league plays out the round.
Vazquez became its third coach of this campaign on Tuesday, replacing Domingos Paciencia after just six games in charge. Vazquez last coached for Celta Vigo in 2006-07.
Vazquez said that Deportivo needed to win eight of its remaining 14 games to avoid relegation. So far, it has won just three of the first 24.
"We need to work harder and better," Vazquez said. "We have a difficult month ahead and we all know it, but we will keep fighting and try to get a win."