Dyche was again without Ashley Barnes, Chris Wood, Johann Berg Gudmundsson and Robbie Brady, who haven't featured since the restart, as well as the departed Jeff Hendrick, Aaron Lennon and Joe Hart.
Jay Rodriguez and Matt Lowton were ruled out ahead of the game, and Jack Cork came off injured in the second half, leaving Burnley to see out the win in a 5-4-1 formation.
The three points moved Burnley to eighth in the table - potentially a Europa League spot - above Arsenal and Everton, and level on points with Spurs.
And Dyche said:"I think it's certainly up there as one of the biggest results I've had as manager here, for the reasons that we're so stretched, we're playing a lot of football, commitment to the cause, but also some of the football was excellent.
"They adapted to the challenge, of not having as big a physical presence up front, because Vyds (Matej Vydra) is more of a mover, a clever mover, and we adapted well.
"Our energy - I've always taken great pride in the fitness of the players, and mental fitness as well, which is important, the stimulus they give each other as a group and collective.
"All those qualities were on show to get what is an excellent result."
Burnley arguably edged the first half, enjoying 53% possession and having five shots to Palace's four, but Roy Hodgson's side penned the Clarets in for periods after the break, without seriously testing Nick Pope.
Dyche said: "We expected that, I said to the group at half-time it was unlikely, improbable they wouldn't come out more on the front foot and get after us a bit more, which they did, so credit to them, but I've said it may times, it's very difficult to dominate a whole Premier League game, they are good sides and will probably have spells in a game, and we had to see it through in a different way second half.
"We scored a great goal, and then we lost Corky, which was a blow to us, and couldn't get Dwight in the game as much - I thought he was excellent in the first half, and Josh Brownhill was excellent all night actually.
"When you think we've got a left back playing left side midfield, Dwighty in an unfamiliar role - although he's a top talent and it doesn't surprise me he delivered a performance - the lads playing game after game, defending, making blocks, the crosses that were dealt with at the end there, Ben Mee scores and makes a brilliant header defending at the other end - all of the qualities that make a team, an effective team."
Burnley earned a return to Europe after 51 years in 2018, and as things stand, would be back in the Europa League, is Manchester City's UEFA ban is upheld.
However, Dyche isn't thinking about that at present, with so many players unavailable: "I think the main thing is that's 13 clean sheets and 13 wins, that sounds pretty good to me. If you get both ends of the pitch right, you'll probably have a decent season.
"It's probably one of those things that's awkward right now because of the player situation, if we had the players with us, I think we could go even stronger to the end of the season, because everyone knows we're physically very fit and sharp, and that's been on show.
"Obviously, having been through it (Europa League), it is very tough, you can't really speak about it at the moment because you've seen how few players we've got.
"That just factually has to be addressed, it's a different ball game.
"That's just common sense.
"The grandeur for a club like us making it into the Europa is considerable, it does cause many challenges, and we will have a shorter break this season because of the Covid situation.
"But the way we've built the mentality here is to keep winning, or attempt to, and the players are open-minded, they'll take each challenge as it comes, and hopefully we'll get some players fit and back as part of the squad at least, but they deserve a massive amount of credit, I can only thank them and remind them how proud I am for what they give to me, themselves and the club."
Dyche again could only field seven of the permitted nine substitutes, but hopes for better news as regards Rodriguez's fitness for Sunday's clash with fellow European hopefuls Sheffield United at Turf Moor: "It's the reality, we've been unfortunate with Jay Rodriguez - he might have a chance on Sunday,
"Lowts I don't think will, he got a stress response to his foot, so that will be unlikely to settle quickly.
"Corky we'll wait and see how bad that is in the next 24-36 hours, but we take on the challenge.
"The next one's tough, but they're all tough, Sheffield United put a lot into their games as well, and we're asking a lot of these players, but they keep delivering.
"There's a superb mentality about what they do."