The Clarets arrived at Goodison Park as heavy underdogs having failed to win a single away game all season, whereas Everton had won all seven of their home matches since the turn of the year.
Ultimately both records were extended, although Burnley more than held their own for the vast majority of the contest before an own goal from Ben Mee and a Romelu Lukaku strike in quick succession took the game away from the visitors.
"It is a tough one to call. You don't want to cry about it but I thought the performance was good. If you were a neutral and didn't know we hadn't won away from home, you wouldn't believe it," Dyche told reporters.
"They are on a very hot run at home but I thought first half we were excellent: we had one cleared off the line, two one v ones and their keeper has made a big save. Second half I didn't think there was that much in the game but the second one (Everton goal) has to be bad luck.
"To have a shot hit your centre-half and it is going wide and then it hits your other centre-half that has got to be bad luck. They have three minutes where they turn up the heat and one of the top strikers in the game at the moment scores a goal. That is the Premier League."
Burnley are one of only two teams in England's top four tiers still waiting for their first away win of the season.