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.

The changes come as part of what the police force has called the "biggest shake-up" in policing in years.
The PSNI said the teams will "deal with community problems" and give "advice".
The first LPTs to launch will be based in Lurgan and Armagh city police stations in County Armagh. They will be supported by four mobile neighbourhood policing teams (NPTs).
The roll-out of all 26 LPTs is expected to be completed by 1 October.
Assistant Chief Constable Stephen Martin said policing in Northern Ireland is in a process of "transformation" due to "significant budget pressures" and new council structures.
He said "normal policing will carry on" and the new teams will respond to calls, conduct investigations, and "deal with community problems".
"They will provide advice and reassurance and spend time learning about what most affects communities in order to work with them to improve safety and quality of life," he said.
The 34 NPTs, which will be based in areas with higher levels of crime and deprivation, are to be set up with the aim of providing "an additional dedicated policing presence, reassurance and commitment to communities where it is most needed".
ACC Martin said restructuring means the number of police stations currently in existence is under review.
"The reality is that nearly all policing services are delivered outside of stations," he added.
"Policing isn't about buildings it is about officers working with the community in the community."

Paul captured global attention after he successfully predicted the results of all of Germany's seven games at the 2010 World Cup, as well as the final.
Each time he opened a box marked with the winning country's flag and ate the mussel inside.
There are now eight contenders in the race to become the next 'psychic' octopus for the Women's World Cup.
One of the those in the running is Lola from France, aged around five months, who lives at the Sea Life Centre in Munich.
Jens Bohn, who works at the aquarium, thinks she is in with a good chance, and he he told BBC World Service that Lola is both "intelligent" and a "quick learner".
He has been training her in preparation for her first test on Friday, when she and the seven other octopuses will attempt to predict the winner of the opening game between Germany and Canada.
But could there ever really be another Paul?
"Perhaps we'll discover that Paul was the only one, the magic one, and the others are just normal octopuses," admits Bohn.
"Nobody knows, but it's worth a try, isn't it?"
Sixteen teams, including England, are taking part in the Women's World Cup in Germany, which runs from 26 June until 17 July.
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.
[NEXT_CONCEPT]
The Police Service of Northern Ireland has started to phase in local policing teams (LPTs).
[NEXT_CONCEPT]
The hunt is on to try to find a successor to the late Paul the 'psychic' octopus.