Striker Sam Vokes deservedly put the hosts ahead just before half-time with a calm finish.
But that lead lasted less than two minutes before Chris Maguire slammed home from close range to earn a share of the spoils at Turf Moor.
"It was one of those strange ones (games) where we have been totally dominant from start to finish in virtually every level of the game," said Clarets boss Dyche.
"Other than the scoreline, I'm delighted with the performance. We had a plethora of chances but we'll have to do that at sometime in the season, dig in and you just get what you can.
"It was a poor goal for us to concede. We'd just scored and given away a soft free-kick which isn't like us and then didn't deal with a ball into the box.
"I thought we played some really good mixed football. It's the most effective way of putting the opposition on the back foot and we have to find that mix of playing long and then getting the full- backs on and stretching teams. Danny (Ings) had a number of chances - he's kept one low and hit the post."
But that lead didn't last until the break as Maguire drew the Owls level, firing home from close range after Burnley failed to clear a Connor Wickham free-kick.
Danny Ings, who had a couple of chances in the first half, hit the post late on and - following QPR's 2-1 victory over Huddersfield - the Clarets slipped down to third place.
"It's a difficult place to come," said Owls caretaker boss Stuart Gray. "They (were) second in the league and a quality side, probably the hardest-working side we've played with two banks of four which close you down and work their socks off.
"Then you've got the boys Vokes and Ings up front and I thought we did well up until the 43rd- minute when we had the opportunity to clear the ball and didn't.
"But credit to my lads they got back into the game. I thought we were better second half, our back four were put under far too much pressure in the first half and our ball retention up front could have been better.
"The brief when I took over was to get out of the bottom three, to keep clean sheets and to score goals which I've done.
"Credit to the lads. Our game plan was to keep them (Vokes and Ings) quiet and we have done."
- Sports & Recreation
- Chris Maguire
- Sam Vokes