Sevilla sacked coach Michel and replaced him with Unai Emery in January and have now won three of their five league games since he took charge. They are 10th on 32 points, five off the European berths.
Croatia midfielder Ivan Rakitic headed the Andalusian side in front after five minutes at the Sanchez Pizjuan stadium although the ball appeared to brush Medel's hand on its way into the net.
The Chilean doubled the home side's lead in the 27th when he turned in a Jesus Navas cross before Riki pulled a goal back for Depor five minutes later.
Medel, a defensive midfielder who rarely gets on the scoresheet, clipped home his second goal in the 40th and Sevilla goalkeeper Beto then pulled off a couple of fine saves in the second half to deny Depor a way back into the match.
The financially-troubled Galician club, who have fallen a long way since winning La Liga in 2000 and reaching the Champions League semi-finals four years later, remain in last place on 16 points.
They were relegated in the 2010-11 season and came straight back up to the top flight but have recently entered administration and appear doomed to second-division football again next term.
"There are a lot of matches left and a lot of work still to be done to pull ourselves up the table," Rakitic said.
"It was very important to win at home and we want to try to win all our games here if we can," he added. "This was a very important step for us."
Leaders Barcelona can pull 15 points clear of second-placed Atletico Madrid with a win at Granada on Saturday in their last match before Wednesday's Champions League last 16, first leg at AC Milan.
Atletico visit mid-table Real Valladolid on Sunday before champions Real Madrid, four points behind their city rivals in third, host another Madrid club, sixth-placed Rayo Vallecano, at the Bernabeu.
Deportivo de La Coruña enjoyed possession for 37.6% of the game.
Deportivo de La Coruña had 12 shots, including five on target.
Nélson Oliveira made the biggest impact of any Deportivo de La Coruña substitute after coming on for Paulo Assunção in the 66th minute, as he took two shots.
The three attempts by Riki were the most of any Deportivo de La Coruña player.