The home defence throughout the afternoon had held out resolutely until Johnson scored in the 49th minute.
Chris Lines drilled in a free-kick from the left, Johnson ran on to the ball past a static defence, and although O's goalkeeper Paul Rachuba managed to parry the header, the Owls defender reacted quickly to stab the loose ball into the net from close range.
The visitors deserved their win in front of a crowd of 6,196, having created a number of chances and only a last-ditch interception by Scott Cuthbert prevented them from taking the lead just before half-time as
Michail Antonio shaped to drill Reda Johnson's cross into the net.
Orient created little although they almost snatched a point deep into added time when substitute George Porter drilled a first-time volley inches wide of the post.
The Owls' win sees them leapfrog Steel City rivals Sheffield United, while at the opposite end of the table the O's slipped closer to the drop zone.