Aaron Ramsey's 12th-minute strike settled a tightly contested match at Turf Moor as the hosts struggled to create clear-cut chances.
Ashley Barnes and George Boyd failed to capitalise when Burnley's best second-half openings came their way, meaning Dyche's men lie two points from safety in 19th position with six games of the season remaining.
Boyd's winner against Manchester City last month stands as the club's only goal in their past six matches.
"There were moments but you've got to take the moments and that's something that we've failed at all season," Dyche said at his post-match press conference.
"[In the] second half we got right on the front foot and went after them much better than we did in the first half.
"You've got to take the chances you create - we created certainly two good ones but you've got to take them at this level, particularly against sides like Arsenal because they don't give you many."
He added: "We've shown this season that we can score goals. We've got to bring that on thick and fast, we know that. There's six big games coming."
Overall, Dyche was pleased with another performance in keeping with the plaudits his players have earned throughout the campaign - meaning the defensive lapse that allowed Ramsey to pounce came as a bitter disappointment.
"The only disappointment of the game, in the bigger picture, was the goal," he said.
"There's an obvious desire from the team - it's well documented and you could see it today.
"There's a good belief in the side and moments of quality that we need to bring to win football matches.
"We've been very good lately at keeping the back door shut but now it's about finding that balance between the two.
"It was a really awful goal. Other than that I thought we defended very well and limited them to a couple of breakaway chances."
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