The Clarets, without a win in seven matches, did not deserve to take anything from the game and have dropped to 19th in the Premier League after Southampton drew 2-2 with Manchester United in Saturday’s 17:30 GMT kick-off.
Palace’s first home victory of the season saw them end a run of eight league games without a win and move up three places to 14th.
“The best side won, simple as that,” Dyche told BBC Sport. “We are in the position where the eye of the tiger has gone. Last season everyone was on it.
“To get the eye of the tiger back, a boxer has to get back in the gym. We’ve got to get back on the training ground and work hard.
“It’s frustrating, they have given so much to the cause but then fear comes in, it’s a tough thing. It plays tricks with you, and players get stretched as individuals.”
The Eagles took the lead early on when James McArthur’s low cross eluded everyone to drift in at the back post, before Andros Townsend added a second with a superb left-footed drive 12 minutes from time.
It could have been worse for Burnley, who had Joe Hart to thank for a number of fine saves, while Palace missed several good chances prior to Townsend’s stunner.
Prior to Saturday’s result, the Selhurst Park season-ticket holders had not witnessed a home win since the final game of 2017-18 – but it was worth the wait.
Given the way Palace started the game, it was difficult to believe they had only claimed two points at home prior to this match.
Roy Hodgson’s side dominated from the first whistle, with Wilfried Zaha and Andros Townsend doing the bulk of the damage.
They deservedly – and perhaps fortunately – took the lead on 16 minutes thanks to McArthur’s intended cross from the left, but they could have gone ahead earlier.
Cheikhou Kouyate was inches away from tapping in a Zaha cross on four minutes, and Hart had to be at his best shortly after when he tipped Max Meyer’s curling effort round a post.
The Eagles missed a spate of chances in the second period – with Meyer, Luka Milivojevic, Zaha, Townsend and Patrick van Aanholt all guilty parties.
It felt like they could pay for their poor finishing, but Townsend’s goal-of-the-month contender made the result safe late on.
“I don’t think we are as bad as our position would suggest, and I think our performance today showed that,” Hodgson told BBC Sport.
“It was a game we needed to win, but though we haven’t played that badly, we have not taken the points.”
