The Owls won a frantic encounter with the only goal in the 33rd minute when a long throw-in was met by giant right-back Johnson and his header sailed across goal into the far corner of the net.
Wednesday had two other openings before the break. Ben Marshall's free-kick was tipped over by Pools goalkeeper Scott Flinders and the Owls midfielder then saw his dangerous ball into the area flash across the face of goal.
But Pools had their chances in a frantic second-half. Antony Sweeney headed wide from Nobby Solano's cross and the Peruvian headed a loose ball over from similar distance.
Pools substitute Andy Monkhouse cracked a low 20-yard drive against the woodwork and a goalbound Evan Horwood header was diverted away by Owls defender Danny Batth.
Wednesday almost got a second on the break in injury time, but Liam Palmer shot wide from a tight angle after skipping round Flinders.