James Chester scored the decisive penalty with Vale's Richard Duffy the only man to miss from the spot.
Over 120 minutes, Vale matched Albion, who again left wantaway striker Saido Berahino out of their side.
The Baggies' Salomon Rondon missed the best chance of the game - hitting the crossbar from close range.
West Brom reached round three of the League Cup courtesy of a penalty shootout after being held to a goalless draw by League One side Port Vale.