The striker scored shortly before half-time, and West Ham never came close to an equaliser for the rest of the tie at the Emirates Stadium.
Here's a look at the teamsheet, via James Olley of the London Evening Standard:
Arsenal's rotated squad took control early, and Welbeck threatened with a weak header after a good cross from Mathieu Debuchy.
The Gunners were comfortable in possession, but it became clear early on that their fringe players and back-ups were lacking rhythm and their passing was sloppy. Chances were rare, and when Welbeck did make a run into the box, Hammers goalkeeper Joe Hart easily intercepted his pass.
James Benge of the Evening Standard was not impressed:
Angelo Ogbonna did well to get in front of Olivier Giroud on a cross from Theo Walcott, who was Arsenal's most dynamic attacking force in the first 30 minutes.
Both teams had minor opportunities late in the half, with Ogbonna scuffing a header and Sead Kolasinac cutting inside but aiming his shot wide.
Walcott should have opened the scoring after 39 minutes, but the speedster somehow failed to hit the target with a wide-open header after a cross from Kolasinac. Three minutes later, Welbeck finally broke the deadlock, firing home after Debuchy nodded the ball across the face of goal.
Per Telegraph Football, it wasn't pretty:
Joe Willock could have doubled the advantage but blasted a shot over the crossbar before the match official called a halt to the half.
Per Arsenal's official Twitter account, an unusual substitution took place at half-time:
The Gunners were happy to control the pace in the second half and, as the visitors were unable to create much, went looking for a second goal to kill the tie.
Giroud had a hopeful header blocked, while at the other end of the pitch, Aaron Cresswell wasted a free-kick from a great position when he curled a shot wide.
West Ham introduced Andy Carroll to try to turn the tide, and the striker clashed with Francis Coquelin soon after, leaving the latter in pain, per the Mirror's John Cross:
Hart brought down Welbeck just outside the box, running into a yellow card and giving away a free-kick from a tricky position. Mohamed Elneny couldn't get the ball through the wall, however.
Carroll just failed to reach a tantalising cross, and Elneny ran out of time after a great run and pass from Willock.
While West Ham needed a goal, the team barely tried to push forward in the final stages. It was a disappointing and uninspired effort from the visitors, while Arsenal lost little energy ahead of the Premier League game against Liverpool on Friday.