Canaries substitute Simeon Jackson netted the winner five minutes from time after Baggies striker Marc-Antoine Fortune had cancelled out Grant Holt's ninth goal of the campaign.
But the visitors' hero was 19-year-old Steer, who produced a series of fine saves - including two in the closing seconds from Shane Long and Joe Mattock.
Lambert said: "I thought his saves were world class. He was terrific. It is only one game but he has a great temperament and a great enthusiasm for the game as well which is important.
"He is a really level-headed lad, which is probably more important than anything - but his handling was brilliant. His two saves (at the end) were exceptional.
"I've never had any fear about playing any of the younger lads. You won't know about someone until you give them an opportunity and, if you don't give them an opportunity, how do you know?"
Victory enabled Norwich to complete a quickfire double success over Albion after they triumphed by the same scoreline in the Premier League clash at The Hawthorns two weeks ago.
Lambert said: "I'm delighted to go through and glad it never went to a replay. This is a hard place to come and again we came up trumps.
"We knew what to expect and that the crowd would be vibrant as they showed a fortnight ago.
"It is well documented the number of late goals we've scored in the last two-and-a-half years and that comes from a great desire within the team.
"They want to try and win games and I always think we will create things."
West Brom head coach Roy Hodgson had sympathy for Albion defender Craig Dawson after his error gifted Jackson his late winner.
He said: "It is one of those no brainer mistakes and I feel sorry for him - and he had not had a bad game up until then.
"Those errors do occur. I am sad to lose because the performance was one where we did not deserve to lose the game, although we didn't really want to go to Norwich midweek for a replay.
"But now we can concentrate on our main priority, which is staying in the Premier League."
