Braga were on the attack from the start, with Ricardo Pinho shooting just wide inside the opening five minutes, before finding the back of the net soon after, rifling the ball in from the edge of the six-yard box.
Portuguese winger Marco Matias had a couple of chances to get the Owls back on level terms before the break, watching a dangerous ball into the box turned behind for a corner before being denied a penalty after going down under pressure in the box.
The visitors made a bright start to the second half as they continued to seek an equaliser, with Stevie May crossing the ball in towards Caolan Lavery, but the Northern Ireland striker was denied by the keeper.
Lewis McGugan did manage to beat the keeper a couple of minutes later, only to see his chipped finish come back off the crossbar.
Wednesday's pressure paid off though as Lavery tapped the ball in from close range after another quick break by McGugan, leaving the Owls level with just over 25 minutes remaining.
The visitors started to look the more likely as the match headed into the final 20 minutes, with a looping shot from Matias almost handing them the lead for the first time in the match.
However, the hosts restored their advantage with just over 10 minutes to go, as Alan volleyed the ball from near the penalty spot following a corner, securing the win for the Portuguese side.