Dyche 'raging' over Arsenal's stoppage-time penalty
Sanchez dispatched a 92nd-minute spot-kick to deny the home side what would have been a well-deserved share of the points.
James Tarkowski was adjudged to have pushed Aaron Ramsey to the ground, enabling Sanchez to score the Gunners' third consecutive stoppage-time winner in this fixture.
The Burnley manager said he was not surprised to be on the wrong side of the decision.
"There is a lot of me inside that is raging. A lot," Dyche told BBC Sport.
"My view is that it was highly unlikely that anything other than a penalty was going to be given.
"But I have got to look beyond that. Our performance today was excellent. [Arsenal] are a fine side and we kept them very quiet, very limited.
"They have turned games in their favour many, many times. The frustrating thing today was we had to turn everything in our favour.
"There was some stuff that was miles out of our control. We can only affect what we can control and I thought the lads did that well overall."
REACTION: 'Soft' Penalty A Kick In The Teeth, insists captain Ben Mee.
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Despite missing a chance to move into the top four, Dyche urged his players to remain focused on the evolution of their playing style.
"I told them we continue to develop and we continue to move forward," he said.
"I think we do [keep improving], both in the manner we are going about it and the effectiveness of our play.
"I have also told them this is the reality. Football is a harsh game and sometimes it hurts you."