Craig 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.
Turner said: "That was probably the most inept performance from a team I have put out for some time.
"It was an embarrassing performance, although as manager I have to take responsibility. I pick the team."
When asked did he pick the wrong team, he replied: "With a scoreline like that and such an inept performance, the answer has to be 'yes'."
After beating a 'big' team like Sheffield United in the league last week, Turner said: "Walsall are a far better team than them at the present time. Sheffield have been struggling at the wrong end of the table.
"This was every bit as important as a league game because of the interest the cup creates and the finance it can bring to the football club.
"In the last 11 minutes when they were three up Walsall started turning on the party tricks and it could have been humiliating."
Walsall boss Smith also praised the performance of young Slovakian under-21 international striker Milan Lalkovic who is on loan from Chelsea until January, and 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.