The 20-year-old midfielder set up the Sheffield side's first before grabbing a double to seal Wednesday's second win of the season.
The Owls, with just two wins from 10 games, almost found a way through in the 10th minute as Brunt whipped in a fierce strike that was pushed around the post by Stephen Bywater.
Wednesday's opener came on 17 minutes when Brunt produced a teasing corner that was met by centre-back Graham Coughlan who finished with a textbook header from six yards.
Coventry danger man Gary McSheffrey weaved his way into the area and his low drive was blocked for a corner before Wednesday went close to grabbing a second as Chris Eagles' stinging 25-yard free-kick was tipped over by the Coventry keeper.
The Sky Blues were back on level terms in the 33rd minute when McSheffrey produced another dazzling run and was stopped in his tracks by David Lucas. Referee Lee Mason pointed to the penalty spot and McSheffrey slammed the ball to the keeper's left.
Brunt fired a free-kick over the top before Wednesday snatched the lead again in spectacular fashion. The midfielder let fly from 30 yards and his left-foot shot zipped past the astonished Bywater and into the top left-hand corner.
Craig Rocastle, recalled by Sturrock in place of Burton O'Brien, was foiled by the Coventry keeper in first-half injury time as Wednesday turned the screw.
The woodwork saved Coventry on 49 minutes as Brunt's free-kick found its way through a crowded area to Lee Peacock who toe-poked against the foot of the post.
Coventry grabbed an equaliser in the 74th minute as Andy Morrell headed in Richard Duffy's short corner before Brunt struck for the second time five minutes later.
Peacock's surge into the area was brought to a halt by defender Andrew Whing and Brunt finished clinically from the penalty spot.