Javier Camunas scored the winner 10 minutes from time to move his side a point above the relegation trapdoor.
Borja Valero had given the Yellow Submarine a 22nd minute lead and, while Sevilla drew level before half-time when Bruno could only deflect Jesus Navas' shot into his own net, the visitors claimed their breakthrough win.
Third-placed Valencia played out a goalless draw at Atletico Madrid who are still to concede a goal since Diego Simeone took charge last month.
Inaki Tejada's first game in charge of Sporting Gijon ended in a 1-1 home draw with Osasuna.
Dejan Lekic scored 13 minutes from time to give Osasuna a share of the points after Carmelo Gonzalez headed a 33rd-minute opener.
Rayo Vallecano came from behind to beat rock-bottom Real Zaragoza 2-1 at the Estadio La Romareda.
Rayo struck twice in the final 15 minutes through Diego Costa and Miguel Michu to snatch victory after Helder Postiga fired the hosts ahead on 33 minutes.
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