The Owls' top scorer took his tally to 10 for the season to complete a comeback victory after Bryan Mbuemo opened the scoring in the first half.
The result takes Wednesday above their rivals and up to sixth in the table, whilst the visitors remain ninth, two points adrift.
The hosts were unbeaten in six league matches at Hillsborough before the start, whilst the Bees had also won six out of the last eight league games.
Both sides were cancelling each other out in an even contest, with neither goalkeeper made to make a save after 25 minutes.
The match felt as though it needed a goal, and Brentford did just that through Mbuemo in the 29th minute.
Mbuemo might have scored just before the goal after his volley went inches wide, but made no mistake from Josh Dasilva's corner as he headed in from close range to put Brentford one goal up.
The hosts looked to bounce back after conceding, with Murphy's low cross again causing trouble for the Bees' rear-guard, but Fletcher couldn't get on the end of it.
Wednesday came out with a point to prove in the second half, as they spurned three golden chances within 10 minutes of the restart.
Morgan Fox's whipped cross found its way to Fletcher in the box, who controlled the ball well on his chest, but his acrobatic effort went wide of the mark.
Then Harris slid Murphy through on goal but the winger waited too long to shoot and his effort was blocked to safety.
The hosts' best chance of the game fell to top scorer Fletcher again, but Brentford goalkeeper David Raya did well to deny his effort at full stretch.
The Owls' pressure paid off in the 69th minute as Fletcher notched his ninth of the season from the penalty spot to level the scores.
Brentford's Rico Henry was adjudged to have handled the ball for the penalty, before the striker stepped up and coolly converted from the spot.
Wednesday's marksman struck again four minutes later as he controlled well from Harris' cross to turn inside and finish past goalkeeper Raya.
Brentford could've equalised in the 90th minute as Said Benrahma broke away from his defender, but his curled effort went agonisingly wide as Wednesday held on for all three points.