The win was Everton's second straight away win in the top flight this season and it lifted them to 14th position on the league log, eight points above the relegation zone.
Idrissa Gana Gueye struck the decisive winner four minutes from time after Palace fought back to draw level twice when Vitaliy Mykolenko and Abdoulaye Doucoure gave Everton leads.
“Obviously topsy-turvy,” said Dyche. “We started so well, with a fantastic goal. They responded with a moment we should have dealt with earlier. They get a soft penalty from our point of view.
“Then really it was a strange game after that. I don’t think anyone really gripped the first half, and second half (Palace) did. I thought they were very good. I think we had to work very hard.
“I think the mentality I’m trying to work with the players on is the belief in finding different ways of winning games, and I think we’re showing that.
“We had to do it last week against Brighton, so nearly won but got a good point. Then coming (to Palace) today.
“Away form here was a big question mark when got here. We’re beginning to change the mentality towards these games. That was on show today.”
Everton will host Manchester United after the November international break.
