The visitors were the masters of their own downfall early on, failing to deal with Neil Clement's corner and allowing Paul McShane to fire in the opener.
And John Hartson made it 2-0 inside 15 minutes from Russell Hoult's long punt.
Kevin Phillips rolled in late on to make it three, before Paul Robinson sliced into his own net in injury time.
Leeds manager Dennis Wise:
"We had opportunities to score but we didn't defend as well as they did; they finished off chances, and unfortunately we didn't.
"I think we're gradually improving and I can't argue with the way they performed in the second half, but the first-half display let us down.
"Overall, I'm disappointed with the way we defended. We had chances to make it 3-3 or 4-3 but we never got close enough to them with our defending. You can't defend the way we did - and they know it."
West Brom: Hoult, McShane, Davies, Clement, Robinson, Kamara (Koren 82), Greening, Chaplow (Carter 81), Koumas, Hartson, Phillips.
Subs Not Used: Zuberbuhler, Gera, Hodgkiss.
Goals: McShane 6, Hartson 15, Phillips 82.
Leeds: Sullivan, Rui Marques, Foxe, Heath (Armando Sa 46), Elliott, Blake, Howson (Derry 46), Nicholls, Lewis, Healy, Moore (Flo 46).
Subs Not Used: Westlake, Lund.
Booked: Foxe, Elliott.
Goals: Robinson 90 og.
Ref: N Miller (Co Durham).