Chelsea's win was overshadowed by the pointed absence of Enzo Fernández, left out of the squad entirely after reports in Argentina suggested he had asked to sit out as he pushes for a move to Manchester City. Xabi Alonso rejected that version of events.
"It was a decision I took," he said. "He's a good professional. He's training well and I have nothing to say about him.
"It was just a decision that we had to take, thinking about the game that we're playing tonight, and we'll see for Sunday [against Brighton]."
The match itself took time to open up: after a goalless first half, Welbeck headed in from close range early in the second period, converting a cross from the right, before Hakeem Odoffin turned a low Estêvão Willian effort into his own net to make it 2-0 with 11 minutes remaining.
Across west London, Rodrigo Muniz scored his first goal in over a year as Fulham cruised to a 3-0 win over their League One opponents, delivering new manager Álvaro Arbeloa his first victory in charge.
Emile Smith Rowe headed the Cottagers in front in the 37th minute, rising to meet a deep Kenny Tete cross that Ryan Sessegnon nodded into his path — with Sessegnon's brother, Steven, playing for Wimbledon on the opposite side. Muniz doubled the lead before the break, meeting another Tete delivery to end a barren run stretching back to the opening weekend of last season, and summer signing Gonzalo García added a third in stoppage time, rounding goalkeeper Nathan Bishop after a pass from teenage debutant Macaulay Zepa.
Fulham made ten changes from Monday's Premier League defeat by Chelsea, but their strength in depth was still too much for a Wimbledon side yet to win a match inside 90 minutes this season.
Zepa, given his first-team debut at the break, made an immediate impact, seeing an early effort turned over by Muniz's audacious attempted overhead kick before later firing narrowly wide of the far post on a lively evening for the 18-year-old.
The results send Fulham into a third-round tie against West Ham, while Chelsea's win books a trip to face Leeds United, continuing what has already been an eventful week at Stamford Bridge on and off the pitch.