Pierre-Emerick Aubameyang made amends for his own goal at the Emirates in December when firing the Gunners ahead with just six minutes on the clock.
The Clarets were caught out when Thomas Partey played through the press, leaving Willian to counter at pace.
The Brazil international fed the ball into the away side's leading scorer, who stepped inside Matt Lowton before beating Nick Pope at his near post.
Dyche said: "It's a tough call on reflection. They started the brighter, but we weren't miles off.
"We let a soft goal in, we pressed really well for our goal, which was part of the intention for our team performance, so I was pleased with that.
"We just looked a little bit jaded, which is to be expected. An awful lot has been asked of them, we're very low on numbers so a lot of them are having to go game after game after game. I can excuse them for being slightly off in the first half.
"I thought we really took the game on in the second half against a really strong Arsenal side. They're a good outfit, they've got good technical players and they can be effective. I'm pleased overall with the way we continued to take the game on."
Burnley weathered a first half storm and were then gifted an equaliser with just minutes remaining in the first half.
Granit Xhaka, who was sent off in Arsenal's 1-0 defeat to the Clarets earlier in the season, tried and failed to play the ball across his own goalmouth.
Chris Wood adjusted his body to guide the ball home for his fifth goal of the campaign.
"Against popular belief sometimes, there's always that Burnley storyline where we wait for a chance and get back in, we don't intend on that," said Dyche.
"Most teams will try and get after sides. I like front foot football and I want my team to play with energy. I thought we got our rewards for doing that."
Wood was denied a winner in the second half by the legs of Bernd Leno while Calum Chambers thumped an effort against the post at the other end.
But the big talking point came in-between. Erik Pieters, who was introduced in the 63rd minute, was given his marching orders for a suspected handball when turning Nicolas Pepe's strike against the bar.
The incident was checked by VAR and, with replays showing that the ball had struck the shoulder, referee Andre Mariner overturned the Dutchman's red card and the decision to award a penalty.
Dyche said: "I thought it was quiet so we got Erik ready and I told him to get involved in as many incidents as you can!
"It's tough because Kante's just had an almost identical situation against Liverpool so they have to find parity somewhere.
"I don't think his arm is unnatural, and Kante's is just the same, but there's a big debate on handball at the minute and I don't like seeing penalties given when it's more ball to hand.
"The big VAR moment is quite clear, but the only gripe I've got is I don't know why Andre Marriner, who is a top ref, has got his card out within a millisecond.
"That was interesting. I was a long way from it but I thought at the time, with our players' reaction, that it didn't hit his arm.
"However, VAR does its job, that's what it's there for. It is there for those moments and it was the right decision."