French forward Wissam Ben Yedder, scorer of the only goal in the Spanish side's 1-0 win in Rome last week, gave Sevilla a two-goal cushion in the tie 20 minutes into the game by pouncing on the rebound after Thomas Strakosha had parried Franco Vazquez's shot.
Lazio required two goals to stay in the competition and they missed a golden opportunity to get back into contention early in the second half when defender Francesco Acerbi sneaked into the area but was thwarted by Sevilla keeper Tomas Vaclik.
The Italians' hopes of a comeback were boosted when Sevilla midfielder Franco Vazquez was sent off for a second yellow card with half an hour remaining and former Sevilla forward Ciro Immobile sent a shot on to the roof of the net.
But Simone Inzaghi's side surrendered their numerical advantage 11 minutes later when Adam Marusic was given a straight red card for striking Roque Mesa in the face.
Sevilla midfielder Sarabia closed out the tie in the 78th minute with a well-worked goal, playing a one-two with Jesus Navas down the right wing and racing into the box to smash the ball inside the near post.
The win eased the pressure on Sevilla coach Pablo Machin after Sunday's chastening 3-0 defeat at Villarreal in La Liga and should boost their confidence ahead of Saturday's game against leaders Barcelona.
Lazio suffered their third straight defeat in all competitions.
(Reporting by Richard Martin, editing by Ed Osmond)
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