Welbeck was accused of deceiving the referee to give the penalty after he fell down in the box, when replays showed that there was minimal contact from AC Milan defender Ricardo Rodriguez.
Borini made a good run before whipping a cross towards Andre Silva, who should have done better than fire into the side-netting.
Xhaka's long-range effort bobbled into the net for Arsenal's second after poor goalkeeping by Milan's highly rated Gianluigi Donnarumma, before Welbeck headed in a third.
Two clubs who have between them capitalised three times in as many seasons on the changed rule that gives the winners of the Europa League a coveted spot in the following season's Champions League group stages.
Atletico, victor of the competition in 2010 and 2012, advanced 8-1 on aggregate.
The first leg will be on Thursday 5 April, with the return taking place on Thursday 12 April.
There was also trouble in the previous round against Spartak Moscow when a police officer died after clashes involving both sets of fans outside the stadium.
Welbeck, hours after being called into the worldwide squad for friendlies against the Netherlands and Italy, sent Donnarumma the wrong way for his first goal since January 7.
Their latest success followed improved performances in the first leg against Milan and Sunday's 3-0 victory over Watford that between them have eased the pressure growing on Wenger and helped Arsenal move on from what he described as a "nightmare week".
A string of injury-hit seasons had threatened to end his worldwide career prematurely, while his place at Arsenal was also at risk.
Lazio looked vulnerable going into their second leg against Dynamo Kiev after being held 2-2 at home but goals from former Liverpool midfielder Lucas and Stefan de Vrij earned them an impressive 2-0 victory.
The Gunners were vulnerable and Milan took advantage to snatch the lead in the 35th minute.
Portugal's Sporting needed an extra-time header from Rodrigo Battaglia to seal a 3-2 aggregate win over Slovakia's Viktoria Plzen after the hosts had forced the additional period with goals by Marek Bakos.
Borussia Dortmund bowed out of the tournament, though, with a tame goalless draw in Austria sealing their fate.