The midfielder netted his 12th and 13th goals of the season late in the first half to settle a scrappy encounter which the Canaries dominated without looking particularly convincing.
Norwich should now fancy their chances of going on to complete a swift return to the top flight, with excellent form behind them and a favourable run of fixtures between now and the end of the season.
Alex Neil’s side had an early shout for a penalty waved away when Tom Lees appeared to bundle over Cameron Jerome in the box but referee Carl Boyeson awarded a corner.
When the ball came in it was only half-cleared and Stephen Whittaker’s well-struck effort was deflected over.
The in-form Canaries continued to look the side more likely to break the deadlock although they were almost caught on the break when Martin Olsson carelessly lost possession on the halfway line.
Lloyd Isgrove drove into the box but the angle was tight and he dragged his shot just wide of the far post.
Caolan Lavery then failed to get a free header on target as the Owls continued to pose a threat on the break but Norwich were generally on the front foot and went close again when Alex Tettey tested Keiran Westwood with another stinging drive from distance.
Whittaker then fired wastefully over the top after being set up by Wes Hoolahan before Norwich took a deserved lead on 33 minutes after conjuring up a goal out of nothing.
Wednesday should really have cleared their lines after a cross from the left was overhit but Jonny Howson forced a hurried punt that ended up picking up Johnson at the back post and the midfielder outjumped his marker to head past Westwood.
Norwich now had their tails up and made it 2-0 in first-half stoppage time when Johnson doubled his tally from a corner.
More poor defending from the visitors saw the ball drop kindly for the midfielder and he needed no second invitation to blast the ball home from just outside the six-yard box via a slight deflection.
Johnson might have completed a swift hat-trick on 49 minutes with a deflected low drive from Hoolahan’s free-kick but Westwood did well to gather the ball cleanly.
The second half developed into a somewhat scrappy affair, with neither side dominating, although Kieran Lee shot just wide on 62 minutes as the visitors gave as good as they got.
Norwich were looking comfortable without posing much of a threat, safe in the knowledge they had the points wrapped up barring an unlikely comeback from a side with nothing but pride to play for.
Howson tried his luck from 25 yards with no success as the game drifted towards its conclusion, with Wednesday having to play the last eight minutes with 10 men as defender Claude Dielna went off with a hamstring injury with all the substitutes used.
Substitute Nathan Redmond went close with a shot across the face of goal – but a third would have flattered the hosts.