Having routed Pecs 85-63 at home in Salamanca on Tuesday night, Avenida were favourites to progress, but it proved to be hard work against the determined Hungarians, who pushed them until the final seconds.
The game was tied at 62-62 when Nicole Ohlde hit a pair of free throws for Pecs with 6:56 left to go.
Le'Coe Willingham went to the line for Avenida and responded with two shots of her own, and after Anna Vajda committed two quick turnovers for Pecs, Silvia Dominguez made it 66-62 to the visitors with a lay-up.
Pecs would not let Avenida get away however, and Kelly Anne Mazzante cut the gap to three, 71-68, with two free throws with 4:09 left to go.
Gunta Basko then nailed a three for the Spaniards, but Pecs hit back as Vajda's lay-up was followed by Szilvia Torok's three-ball, cutting the lead to one, 74-73, with 1:57 left to go.
Avenida missed a couple of opportunities before Vajda fouled Donnette Snow, who hit both free throws to open up a three-point lead.
Mazzante missed a three-pointer for Pecs and Avenida made her pay as Isa Sanchez went to the line and hit both her free throws to open up a five-point lead, 78-73, 27 seconds from the end.
Ohlde hit a quick jump-shot, but the hosts could not get back in it as the teams exchanged fouls, Anke De Mondt ultimately icing the win from the charity stripe with 10 seconds left.
Willingham paced Avenida with 20 points, but Snow was the star of the show with 17 points and 14 rebounds.
EuroLeague Women All-Star De Mondt finished with 18 points.
Pecs used a balanced attack as Ohlde registered 15 points, Dalma Ivanyi had 14, Vajda 13, Torok 12 and Mazzante 10.