Welbeck, starting for the first time this season after three substitute appearances since recovering, struck twice in the opening 22 minutes and set up England team mate Theo Walcott for the first of his three goals as Arsenal killed off the fourth-round tie before halftime.
"We were very dynamic, explosive and overall we had a good consistent performance over 90 minutes and everybody played well," Wenger, who was forced to watch from the stands because of a four-match touchline ban, said.
"Welbeck has been out for such a long time, I didn't expect him to score straight away and it shows the desire is there.
"He was very sharp because he has been working very hard. He has gone through some tough moments mentally.
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"But his resilience was absolutely marvellous and he was rewarded tonight, so we are happy."
Arsenal, who gave run-outs to young midfielders Jeff Reine-Adelaide and Ainsley Maitland-Niles, are in the last 16 of the Cup for the seventh consecutive year.
It was a chastening night for Southampton three days after they had beaten Liverpool at Anfield to reach the League Cup final where they will play Manchester United.
Manager Claude Puel made 10 changes to his side and paid a heavy price with Arsenal having far too much strength in depth.
"I am responsible for the defeat," Puel said.
"We had nine players not available and I had to protect some players. It is not possible to play every three days with the same players and with some players with injuries.
"It was a good lesson, they are a great team."
(Reporting by Martyn Herman; Editing by Clare Fallon)
