Franco Vazquez put Sevilla ahead at Rayo in the 15th minute after each side had goals correctly ruled out for offside without using VAR, and Silva stretched their advantage with two strikes from inside the area before halftime.
Silva, who joined AC Milan from FC Porto last year for 38 million euros (34.08 million pounds) but scored only two goals in Serie A and moved to Sevilla on loan this month, completed his hat-trick in the 79th minute with another close-range strike.
The goal was originally ruled out for offside but referee Antonio Mateu Lahoz eventually awarded it after consulting the video assistants. He also used the technology to give Rayo a penalty from which Adri Embarba scored a consolation goal.
Silva, 22, has scored more goals in one game in La Liga than in 24 appearances in Serie A and became the first player since former Barcelona striker Romario in 1993 to hit a hat-trick on his La Liga debut.
Sevilla provisionally lead Spain's top flight on the opening weekend on goals scored, with Barcelona below them after beating Alaves 3-0 on Saturday.
La Liga debutants Huesca announced their arrival in the Spanish top flight in style on Sunday by grabbing a 2-1 win away to Eibar in their first game of the season, powered by two impressive strikes from midfielder Alex Gallar.
Gallar danced his way past three defenders before sticking the ball through the legs of Eibar goalkeeper Marko Dmitrovic to score Huesca's first ever goal in La Liga in the fifth minute of a match between the two smallest clubs in the league.
Gallar followed that spectacular goal by unintentionally hitting their second from a long-range free kick in the 40th as his attempted cross went beyond its intended target and flew straight into the net, catching Dmitrovic by surprise.
Argentine midfielder Gonzalo Escalante converted a low cross midway through the second half to pull back a goal for Eibar, who are in their fifth consecutive season in Spain's top flight despite coming from a town with a population of 27,000.
Huesca, from the north-eastern city of the same name with little more than 50,000 inhabitants, finished top of Spain's second tier last season. They had to withstand some frantic late attacks from Eibar but held on to claim a historic victory which took them joint top of the standings on the opening weekend.
(Reporting by Richard Martin; Editing by Clare Fallon)
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