But Boro were almost made to pay for their sloppy finishing as Derby made a game of it in the last quarter.
Substitute Hossam Ghaly showed glimpses of his undoubted talent and the Rams had two great chances to snatch a point in the dying moments.
First, Earnshaw headed wide after Ghaly's fine crossfield pass caused havoc in the home defence.
And Miller then picked the wrong option and ended up shooting tamely at Mark Schwarzer following a swift counter-attack.
Middlesbrough manager Gareth Southgate: "We only got the one goal and then in the last 25 minutes of the game we subconsciously decided to sit on our lead.
"We had a couple of chances to finish the game off but they kept plugging away and only being one goal ahead gave them the chance to get back into it.
"We didn't really play in the second half. Had we got a second goal I think we would have ended up winning quite comfortably but we held on and it was an important win because we have some difficult games to come."
Derby manager Paul Jewell: "In the first half we were rubbish but in the second half we passed the ball well.
"We controlled the last 25 minutes and could have snatched a point. We had two great chances and I think they were glad to hear the final whistle.
"We didn't start the game well and didn't play the conditions. Rob Earnshaw should have done better with the chance he had but I'll never have a go at a player for missing chances.
"Hossam Ghaly ran the game when he came on. He was terrific and showed what a good player he is."
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