Visitors Almeria almost blew a four-goal buffer as second-tier outfit Betis struck three times during the final 10 minutes at the Estadio Benito Villamarin.
However, the La Liga outfit managed to hold on for victory in the round-of-32 clash in Seville and they will now take four priceless away goals back to the Estadio de los Juegos Mediterraneos for the return leg on December 16.
Almeria could not have dreamed of a better start against Betis, with Spanish left-back Mane and Thai forward Teerasil Dangda both scoring in the opening five minutes.
Mane managed to find the back of the net from outside the box in the third minute before Dangda beat Betis goalkeeper Dani Gimenez in a one-on-one situation two minutes later.
Quique made it 3-0 on the half-hour mark, glancing a header into the far corner of the net, and the result was seemingly put beyond doubt in the 61st minute as Edgar headed a ball into the ground and beyond Gimenez.
But Betis refused to surrender, substitute Ruben Castro pulling a goal back 10 minutes from the end.
Betis' comeback continued six minutes later as Polish defender Damien Perquis volleyed the ball into the net from a corner.
The home side had the last say of the match in the first minute of injury time, substitute Jorge Molina heading home powerfully from another corner.
Getafe had no such trouble at home to fellow top-flight side Eibar, triumphing 3-0 thanks to three second-half goals.
Pablo Sarabia bagged a brace, while Alvaro Vazquez came off the bench to score a goal of his own at the Coliseum Alfonso Perez.
Meanwhile, Moroccan defender Issam El Adoua equalised in the 68th minute as Levante earned a 1-1 draw at Albacete.