The Championship leaders made a bright start but were reduced to 10 men shortly before half-time after Jerome Thomas pushed Jack Wilshere in the face.
Vela then came off the bench during the second half and soon created a goal as his powerful shot from the edge of the area was by tapped-in by Sanchez Watt.
The 20-year-old Mexican added a second himself when he latched onto the rebound from Mark Randall's chip for an easy finish.
Arsene Wenger relied on youth for the tie, with his side having an average age of just over 20, while also bringing in experienced defenders Philippe Senderos and Mikael Silvestre.
Senderos, who was playing his first match since rejoining the club from a loan spell at AC Milan, missed an early chance when he headed over at the near post from a corner.
SavesWith Watt and French teenager Gilles Sunu Arsenal certainly pace to burn in attack, while Wilshere and Aaron Ramsey were the creative force in midfield.
Polish goalkeeper Wojciech Szczesny made the first meaningful contribution on his debut when punching a corner clear at the near post and he was stretching in the 15th minute as Robert Koren's 20-yard effort flew just wide.
Watt came within inches of opening his Arsenal account when he stabbed Wilshere's cutback goalwards, only for the ball to cannon off the inside of the post and into Dean Kiely's grateful arms.
At the other end, West Brom midfielder Simon Cox became the next to test Szczesny with a low drive and the Pole produced another fine stop to turn Koren's shot away.
The youngsters were rattled and Kerrea Gilbert was forced to hack the ball off the line after a corner caused panic in the Arsenal six-yard box.
On 31 minutes, Wilshere made space for himself with another positive run before striking a 20-yard shot just off-target.
West Brom swung a free-kick into the area which Szczesny dropped under pressure at the far post from Leon Barnett but fortunately Kieran Gibbs was on hand to clear.
ThreatWest Brom continued to threaten on the break, with Thomas' run down the left forcing Szczesny to make an important block.
However, the 26-year-old former Gunner was shown a straight red card in the 37th minute after he clashed with Wilshere then put his hands in the young star's face
Cox squandered the first real chance of the second half when he headed over while well placed in the six-yard box.
Wenger then made a change, bringing on Vela and Randall to replace Sunu and Francis Coquelin.
A moment of quality from Vela finally opened the door for the deadlock to be broken.
The 20-year-old showed great close control to create space for a shot from the edge of the penalty area, which Kiely palmed wide, and Watt followed up to smash in his first senior Arsenal goal.
The Mexican then got added his own name to the scoresheet on 75 minutes.
Randall's 20-yard chip came back off the underside of the bar and Vela was alert enough to get in between Barnett and Kiely before knocking the ball over the line.
The Baggies missed the chance to pull a goal back with 10 minutes to go when substitute Chris Wood fired wide at the far post.