Headed goals from Danny Ings after 33 minutes and Sam Vokes five minutes later won the game but Wednesday created a nervous finish for the Clarets after pulling a goal back on 70 minutes through David Prutton.
The result leaves the home side still searching for their first point of the season after last week's defeat at QPR while the Clarets are undefeated after an opening draw against Bolton.
It was the Owls who started the brightest, with Michail Antonio twisting and turning his defender before pulling the ball back to the front post where Jacques Maghoma mistimed a great chance and fired wide of Tom Heaton's goal on nine minutes.
Burley came back with Keiran Trippier playing a ball over the top of the defence to Vokes who volleyed his shot wildly over the bar whilst the Wednesday defence protested to the referee about a suspicion of offside.
From then on Sean Dyche's side started to take control, with a deflected Dean Marney shot landing at the feet of Vokes in the box but the striker put his shot straight at Chris Kirkland.
The Clarets took the lead in the 33rd minute after Trippier played Ross Wallace down the right hand side. Wallace took his time and lofted the ball onto the head of Ings who nodded it past a helpless Kirkland.
It didn't take long for Burnley to double their lead, with Trippier beating his defender and whipping a ball to the back-post which was met by Vokes, who headed it back across goal to make it 2-0 after 38 minutes.
Dave Jones' men came out the quickest after the restart, with half-time substitute Chris Maguire somehow finding his way past two players in the box before his shot was parried away by Heaton but the ensuing goal-mouth scramble was eventually cleared.
The home side then piled on the pressure, with Maghoma stealing the ball from Trippier before finding Austrian striker Atdhe Nuhiu, whose header went just wide with half the ground already celebrating a goal.
Wednesday eventually found a breakthrough with 19 minutes to go as right-back Liam Palmer put in a cross for Prutton to attack and his diving header flew past Heaton.
The Owls pushed for an equaliser, with a half-chance coming to Maguire with ten minutes to go, but were almost caught out from a corner after Burnley substitute David Edgar glanced a free header wide.