After a 1-1 result in the first leg in Portugal, 90 minutes could not separate the teams at the Stadio Friuli Tuesday night as Pablo Armero's 25th-minute header for the hosts was cancelled out by another header, from Ruben Micael, in the 72nd-minute.
Extra-time produced no more goals and so the contest went to penalties, with Braga winning 5-4. Brazilian midfielder Maicosuel's miss for Udinese with his team's third kick proved decisive as Micael kept his nerve with the 10th spot-kick of the shoot-out to carry the Portuguese side through.
Elsewhere, Anderlecht left it late before sealing a 3-2 aggregate success against Cypriots AEL.
The Belgians lost the first leg 2-1 and were going out heading into the last nine minutes of tonight's return - until goals from Dieumerci Mbokani and Oleksandr Yakovenko gave them a 2-0 win which took them through.
Joining them in the lucrative group stages are Dinamo Zagreb, who won 1-0 in Maribor to seal a 3-1 aggregate success.
Portuguese defender Tonel got the only goal of the game in the 12th minute for the visitors, with a bad night for his compatriots capped with the sending-off of Soares Arghus five minutes from time.
BATE Borisov drew 1-1 at Hapoel Kiryat Shmona to make sure of a 3-1 aggregate victory.
Laszlo Lencse's free-kick gave the Israeli side hope with a goal midway through the second half but their hopes of progress were snuffed out by Aleksandr Pavlov's equalizer four minutes into injury time.
The fifth team to progress Tuesday night were Malaga. The Spanish side's 2-0 advantage from the first leg proved enough to carry them past Panathinaikos following a goalless draw in Greece.