Striker Sam Vokes opened the scoring for Burnley, who beat Crystal Palace last time out, in the 19th minute following good work from former West Ham winger Junior Stanislas. Danny Ings then capitalised on a Danny Shittu error just before the hour to add a second while Liam Trotter had an effort cleared off the line for the Lions late on.
And Dyche, who made 69 appearances for Millwall between 1999 and 2002, is calling for more of the same from Burnley, who are now four points off the play-off places in seventh.
"It is good fun climbing the table; I wish it was easy enough week in, week out. You've got 11 each week out to get you but the feelgood factor is there," said Dyche.
"I want us to win as many games as possible and who knows where that can take us? That's the challenge, quite simply. We're in the hunt right now but we can only take it one game at a time."
Defeat was Millwall's second in three at home in the Championship and boss Kenny Jackett admitted the Lions' usual spirit was not on show.
"The result didn't please me. We didn't do enough to get something out of the game. Burnley showed more hunger and desire for the game than we did and that's disappointing," said Jackett.
"We started the game poorly, on the wings they got in behind and got off crosses far too easily. The first goal is a prime example. He [Stanislas] went past two to deliver for Vokes to score.
"We didn't defend well, not with everything we've got, not for our life, which we do when we're playing well.
"Going forward, take the start of the second half, we spread the ball across the face of goal but didn't get on the end of it. We needed that goal to go and lift us and lift us again. We changed formation in the last half hour to free up the centre but we couldn't do enough."
- Sports & Recreation