The Owls – who will start the Championship season on minus 12 points after breaking spending rules – won the shoot-out 4-2 as Josh Gordon and Alfie Bates missed for the Saddlers.
It took until the brink of the interval for either side to threaten as Walsall striker Gordon's promising run ended with a shot straight at Dawson.
Wednesday improved after the break, sparked by wingers Harris and Matt Penney.
Harris curled a decent opening over from 15 yards before substitute Penney almost made an immediate impact with a 20-yard effort that Walsall goalkeeper Liam Roberts pushed aside.
The visitors upped the pressure late on but, although Massimo Luongo poked wide from 10 yards and Penney headed a great chance over from Brown's centre, it mattered not.