With Atletico striker Radamel Falcao without a goal for the third straight game, Turan stepped up in the 61st minute to ensure the hard-earned win for Diego Simeone's team that keeps it three points behind pacesetter Barcelona.
The Turkey midfielder volleyed in from substitute Jorge "Koke" Resurrecion's cross after Diego Costa had started the move that succeeded in unlocking Granada's defense.
Granada had outplayed the visitors at Los Carmenes stadium, coming close through Gabriel Torje when he hit the post in the 35th.
Atletico resisted with a man down from the 71st when Mario Suarez received his second yellow card.
"They beat us in every facet in the first half. In possession, in intensity of play - in everything," said Simeone.
"The presence of Diego Costa in the second half helped us create more chances of controlling the ball."
Atletico's 10th victory in 12 rounds ensured it would arrive to the Spanish capital derby in two rounds ahead of third-place Real Madrid in the standings.
Jose Antonio Reyes netted a double and set up one of defender Federico Fazio's two goals for Sevilla in his best performance since returning to his boyhood club last season.
Sevilla could not have asked for a better start at home, jumping ahead just 12 seconds in when Betis goalkeeper Adrian San Miguel gifted the ball to Reyes, who blasted it into the top corner.
"It truly was a complete night, for Sevilla, its fans, and for me," said Reyes. "I don't think I have ever scored such a quick goal."
Its rival rattled, Sevilla poured on the pressure and Fazio met Ivan Ratikic's free kick to turn it beyond the `keeper with a header in the fifth.
Reyes took his second goal in the 32nd by slipping in from behind the flatfooted Nelson Tomar to finish off the rebound of Alvaro Negredo's header that had crashed off the crossbar.
The former Arsenal forward then crossed for Fazio to thump another header just inside the post two minutes before halftime.
Ruben Perez left Betis short-handed in the 62nd after getting two yellow cards in a six-minute span. Striker Ruben Castro pulled one back for Betis four minutes later, before Rakitic capped the demolition in added time.
"I have never seen anything like that first half, it was spectacular," said Sevilla coach Michel Gonzalez. "I take my hat off to all my players."
Betis remained in sixth place, with Sevilla one point behind level with Valencia.
Levante, meanwhile, moved up to fourth place and into the Champions League places after winning 2-0 at Deportivo La Coruna in an eventful match in which both teams finished with 10 men.
Earlier, Obafemi Martins gave visiting Levante the lead in the 40th minute when he latched onto a long ball, dribbled past goalkeeper Daniel Aranzubia and rolled in his fifth goal in eight league appearances.
The contest appeared to have swung in Deportivo's favor when Pedro Lopez was sent off for his second booking for handling the ball in the area.
But Ivan "Riki" Sanchez missed the resulting 55th-minute penalty and Aranzubia was sent off for fouling Martins in the 69th.
Having already made all three substitutions, Deportivo used midfielder Alejandro Bergantinos as stand-in goalkeeper, and Levante forward Jose Barkero sealed the win with a long-range strike in the 82nd.
"We are very happy," said Levante coach Juan Ignacio Martinez.
"(But) we have to acknowledge what happened. Our rival had to substitute both center backs and then tried to go for the game by making a third change. The penalty could have made it 1-1 and the sending off of their `keeper conditioned the match."
Elsewhere, Getafe scored twice in four second-half minutes to edge out Valladolid 2-1, while Mallorca broke a six-game losing run by drawing 1-1 at Celta Vigo.
On Saturday, Barcelona beat Real Zaragoza 3-1 to extend its club-record start, while Madrid thrashed Athletic Bilbao 5-1. Valencia downed Espanyol 2-1, and fifth-place Malaga drew 0-0 at Osasuna.
Rayo Vallecano visits Real Sociedad on Monday.
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