Leading 1-0 from the first leg, Arsenal trailed 1-0 at half-time and survived an Antonio di Natale penalty miss with the score at 1-1 before Theo Walcott scored to lift the pressure.
Twice former European champions Benfica joined them with a 3-1 win over Twente Enschede and Olympique Lyon made it 12 appearances in a row by holding out for a 1-1 draw at Rubin Kazan in the play-off second legs.
Czech champions Viktoria Plzen qualified for the first time, beating FC Copenhagen 2-1, and BATE Borisov completed the line-up by knocking out Sturm Graz.
The five completed the line-up for Thursday's group stage draw in Monaco.
Trabzonspor were named by UEFA as Turkey's representatives earlier on Wednesday after Fenerbahce were pulled out by the Turkish FA over the country's match-fixing scandal.
Arsenal's 3-1 aggregate win was a relief for manager Arsene Wenger after a difficult start to the season, with injuries adding to the departures of Cesc Fabregas and Samir Nasri.
"It's relief and pride as well," he told Britain's ITV . "We had our backs to the wall but there was only one team on the pitch in the second half.
"We showed great belief in the way we want to play football. It was very important tonight, not for financial reasons as many people have been saying, but more because we want to play at the top level."
Arsenal were over-run for much of the first leg and fell behind when Di Natale, who had already hit the post, met Giampiero Pinzi's cross with a looping header in the 39th minute.
The visitors levelled 10 minutes after the break when Gervinho tricked his way to the byline and pulled the ball back for Robin van Persie to score, only for Udinese to be awarded a penalty for handball minutes later.
MISSED PENALTY
It proved the turning point as Di Natale, top scorer in Serie A for the past two seasons, saw his effort met with a flying save by Wojciech Szczesny.
Walcott then settled the tie when he broke clear after a neat one-two with Bacary Sagna and scored clinically in the 69th minute.
Benfica, starting with 11 foreign players including eight South Americans and boosted by a 2-2 away draw last week, scored three times in the second half with two from new Belgian signing Alex Witsel and one from Brazilian Luisao.
Costa Rican Bryan Ruiz replied for Twente.
Lyon's progress was far less comfortable at Rubin Kazan where Bebras Natcho volleyed in Obafemi Martins' pass with 13 minutes left, leaving the Russians one goal from an away goals win.
But they were left deflated after Bakary Kone was first to an inswinging corner to head the equaliser three minutes from time and send Olympique through 4-2 on aggregate.
Plzen, 3-1 up from the away leg, had a fright when Costa Rica midfielder Christian Bolanos gave visitors FC Copenhagen afirst-half lead.
Marek Bakos knocked home Petr Jiracek's cross from close-range to level in the 67th minute, his sixth goal of the qualifiers, and Michal Duris struck deep into stoppage-time to complete a 5-2 aggregate win.
BATE Borisov, held 1-1 at home by Austrian champions Sturm Graz, looked more comfortable playing away as Aleksandr Volodko put them ahead nine minutes before halftime
Serbian Marko Simic, one of only two foreign players starting for the Belarus champions, made the tie safe with the second in the 70th minute as his side qualified for the second time.
