The visitors saw a first-half goal from Josh King ruled out with Philip Billing deemed to have handled Andrew Surman’s corner into his path.
Eddie Howe was then left rueing his luck in the second half when after Matej Vydra’s opener, the Cherries appeared to equalise through Harry Wilson.
But VAR official Chris Kavanagh pulled play back for a handball by Adam Smith, chalking off the Cherries goal and awarding Burnley a penalty.
Jay Rodriguez converted and then Dwight McNeil added a third as the Clarets made it four wins from their last five in the Premier League.
“The first one, I think, is a hard one for them to take, but, inevitably, it does lead to a goal and he does move his shoulder towards the ball, and it does hit his arm,” said Dyche.
“So if you look at the facts of it, you’d consider that is too much, because it led to a goal.
“Although it’s a tough one for them, I think it’s probably the right decision.
“The second is a different kind of disappointment for them, because they think they’ve gone up the other end and scored.
“But Eddie is an honest fella…I thought it was a penalty, as soon as I saw it, I was a defender, and the wind was swirling, it was really hard to play for both teams, but his arm came into an unnatural position, as they call it, and it definitely hits his arm.
“Jeff (Hendrick) is just behind him as well, so if you put those three things together, I think it’s a penalty.
“If he doesn’t stick his arm out, Jeff, in theory, can put it back in the box and we could score from it.
“That’s a definite one for me, hard for them to take, but a definite for me.”
Dyche felt his side could have had another penalty in the opening period when the ball bounced up and struck the arm of Steve Cook in the Cherries box.
“I will say, it’s really tough, having been a centre half, the lad Cook first half, I’ve seen it back, he has a little swipe at it, in my opinion, so that’s a tough one as well,” said the Clarets boss.
“So, a strange, tough day for VAR. That’s the toughest it gets though, if I’m part of their side, the staff, the players, the fans, that’s a tough one but I don’t think that will happen many times when you have a decision and they go and score 45 seconds later or so.
“You think you’ve scored and you get one against you.”
Despite the big VAR debate, Dyche felt his side were well worth their victory that continues their match up the Premier League table, the Clarets now eighth ahead of Sunday’s games.
“It was a funny game, VAR will dominate the key points, but I thought Bournemouth were very good first half, I like Eddie’s teams and they didn’t look like a side having the run they’re having,” he said.
“The wind helped, on the sideline it was really strong and helped them build momentum, but they were still better than us. First half we didn’t get to grips with them.
“Second half it was the opposite, but I thought we looked stronger in everything, our defensive shape, the details, and it was a high chance count, 17 on goal, 10 on target – if you get 10 on target you normally score.
“That being said, at 2-0 we had two or three unbelievable chances to finish the game off and didn’t take them, but we get the third and the game is done.”
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