Facing an uphill task from the first whistle following a 2-0 home defeat a week ago, Sevilla made the perfect start to the game when Alberto Moreno delivered an inviting cross for Jose Antonio Reyes, who finished confidently to break the deadlock in the 19th minute.
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The visitors continued to dominate possession after the early goal but a determined Betis defence restricted the chances on offer before the break.
However, the game opened up early in the second half, with Stephane Mbia coming close to doubling Sevilla's lead on the night when he forced Antonio Adan into a save.
Unai Emery's side were forced to wait until the 75th minute before drawing level and it was Moreno who provided the assist once again as Carlos Bacca arrived in the box to drill home a cross from the full-back.
Both sides pushed for a late winner but a lack of clear chances meant that extra time was needed to separate the rivals.
Adan was called into action in the first extra period to keep out Kevin Gameiro's curled effort and the Betis goalkeeper then responded quickly enough to push away Bacca's header just seconds later.
Chances proved to be rare either side of the break in extra time, with both sides being reduced to trying their luck from distance, and the lack of a clear opportunity ensured that penalties were needed.
Vitolo stepped up first in the shootout for Sevilla but Adan guessed correctly to make a smart save.
Neither goalkeeper could keep out the next six penalties but Beto responded to save Cedric Mabwati's spot kick to level the scores.