Victory lifts the home side into the top 10 while the Owls - who had top scorer Fernando Forestieri sent off - slip to sixth, ending a seven-match unbeaten run.
The hosts had the first chance of the match after 10 minutes when Greg Cunningham’s left-wing cross picked out fellow full-back Calum Woods, who struck just wide from 12 yards.
And they threatened again in the 22nd minute when Garner struck from 25 yards but Keiren Westwood held the dipping effort.
The Preston pressure continued a minute later when Adam Reach whistled a strike from 20 yards inches over the bar and the hosts were successfully silencing the vast away support.
Were it not for Westwood, Preston would have taken the lead after 27 minutes, but the keeper sprung low to his right to save Garner’s header from Paul Gallagher’s free-kick.
Sheffield Wednesday began the day as the Championship’s top scorers but the chances continued to fall for Preston as another Gallagher free-kick made its way to Tom Clarke on 31 minutes but he thrashed the ball wide when unmarked from eight yards.
The Owls’ best chance of the half came after 43 minutes when Barry Bannan whipped in a superb free-kick, but Tom Lees could not get enough of a head on it from six yards.
Preston began the second half on the front foot as Woods’ cross from the right after 51 minutes just evaded Garner, who claimed Lees had pushed him, before Alan Browne fired over from the left edge of the area.
Smart play on the edge of the area released Gary Hooper for the visitors, who flashed a low ball across the six-yard box that nobody was there to meet.
Ross Wallace’s free-kick on the hour mark sparked panic in the Preston area and the ball eventually fell to Daniel Pudil, who volleyed high over the bar from 12 yards.
The crucial moment arrived on 66 minutes when Wednesday’s top scorer Forestieri, who had already been booked for diving, tried to handle the ball into the net and received a second yellow card.
Preston tried to make the most of their advantage and Reach attempted a hopeful long-range strike after 70 minutes that was easily held by Westwood.
But three minutes later they made the extra man count. Woods’ left-wing cross found Garner at the far post and his bullet header from six yards gave Westwood no chance.
The visitors pushed for an equaliser and Lucas Joao looked certain to tap home Hooper’s cross after 83 minutes, but a brilliant last-ditch tackle from Tom Clarke condemned Wednesday to their first defeat in eight league games.
Wednesday manager Carlos Carvalhal was unhappy with the red card shown to Forestieri. He was booked first for diving, then given a second yellow card for handball.
“Forestieri received the ball in a good position and the defence blocked him and the referee gave him a yellow card, so the first yellow is not correct,” Carvalhal said.
“The second yellow seems correct. Even though he doesn’t touch the ball he’s tried to put his hand on the ball and the referee therefore was correct.
“The first yellow card is unbelievable because it was a clear foul to us and he gives a yellow to Fernando. If he doesn’t go down he will be in front of goal in the area with a very clear chance to score. For me he didn’t dive in that moment because if he did it was not a good choice. He blocked him.
“I am happy with the attitude of our players, they tried to win the game, but we are unhappy with the score of course.
“The fans were amazing again, the game was balanced, we both created chances and in the second half we looked as though we would score but the red card changed the game. Even with 10 players we took risks and the players fought until the last second to try and win the game.”