Lawrence, the former Republic of Ireland international, collected a pass from Bobby Grant in the 53rd minute and rifled a rising 20-yard shot past Richard O’Donnell.
Shrewsbury, who will now travel to Preston in the second round, played most of the first half on the front foot. Leading scorer James Collins fired their best chance wide from close range after being set up by a clever pass from James Weslowski.
Grant’s 30-yard free-kick forced a save from O’Donnell before Collins and Nat Knight-Percival were narrowly off target with headers.
Walsall, without injured nine-goal top scorer Tom Bradshaw, carried little threat going forward and rarely suggested an equaliser was likely after Lawrence gave Shrewsbury the lead.
Romaine Sawyers headed straight at goalkeeper Jayson Leutwiler and Ben Purkiss lifted a late chance over the bar from close range.
Walsall’s cause was not helped by an 82nd-minute red card for midfielder Adam Chambers for lunging at Knight-Pervical just inside the Shrewsbury half.