Westcarr opened the scoring with a tap-in goal on 29 minutes following a mistake in the Shrewsbury defence and his second came in the 59th minute after a shot from Milan Lalkovic - on loan from Chelsea - rebounded to him.
Then Westcarr set up Romaine Sawyers to smash in the third in the 73rd minute, while Shrewsbury's best effort was a drive from substitute John Marsden which home keeper Richard O'Donnell beat out late in the game.
"It was a great centre-forward performance and he had a couple of opportunities to get his hat-trick but tried to put other players in instead," said Smith.
"I thought it was a good performance by the team throughout the game and we scored some good goals. The players worked really hard and wanted to score more goals.
"The balance and tempo was good today. You are always concerned when you haven't scored more than one goal after the possession we had in the first half, and we expected Shrewsbury to come at us in the second half, but the whole team defended well."
Smith also praised the performance of young Slovakian under-21 international striker Lalkovic, who is on loan until January.
He said: "That was the best he has played for a while, but this is his first season in the league. He showed his pace and cleverness and linked up well with other players."
Walsall hadn't won a home FA Cup tie since November 2007 - when they beat Shrewsbury 2-0.