Everton, however, had the final say thanks to a late penalty kick from Belgian international Kevin Mirallas.
Celebrating the 125th anniversary of Goodison Park, the Toffees made the ideal start to the afternoon and took the lead with just 34 seconds on the clock through summer signing Sandro.
Dominic Calvert-Lewin and Tom Davies combined following a long kick by goalkeeper Jordan Pickford to send the striker through on goal, allowing him to round Sevilla stopper David Soria and guide the ball home.
Sevilla took a while to recover from their disastrous start, but once they did, they began to dominate the first half, coming closest to an equaliser when Wissam Ben Yedder's strike was hacked off the line by Leighton Baines.
Everton were second best for much of the second half, and the Spaniards netted a deserved leveller just before the hour mark when Banega was given too much space around the six-yard box, allowing the Argentine to rifle a shot beyond Pickford's reach and into the bottom corner.
There was a measure of controversy about Sevilla's second, which came from the penalty spot after former Manchester City forward Nolito was fouled on the edge of the area.
Referee Mike Jones erroneously pointed to the spot, and Banega drilled the ball past Pickford from 12 yards.
Although Everton remained under the cosh, Mirallas was a live wire following his second-half introduction, and his deft turn in the box in the 78th minute drew a foul from his marker and earned his side a penalty.
The Belgian stepped up himself and unleashed a tame effort from the spot, but it proved just enough to beat Soria in the visitors' goal.
Both sides had late opportunities to win it, with Mirallas hitting a screamer just beyond the far post in the dying stages and Jesus Navas rattling the woodwork at the other end moments later.
There was still time for a long-rage strike from Ademola Lookman to draw an impressive save from Soria in the final minute, but the match ended in a stalemate.