The struggling Railwaymen were also helped on their way to victory by the 67th minute dismissal of Swindon striker Billy Paynter, after he had headbutted home midfielder Michael O'Connor.
Swindon were the more dangerous side in a scrappy first half with Simon Cox forcing a near-post save from Stuart Tomlinson and Paynter going close when he stabbed Anthony McNamee's cross into the sidenetting.
Crewe took the lead when Joel Grant was fed outside the box and his powerful drive could only be parried by Phil Smith. Miller nipped in to sidefoot the ball home through a crowd of players for the all-important goal.
The young striker came close to grabbing a second early in the second half when he lobbed Hasney Aljofree, but keeper Smith was off his line to block the striker's volley.
Swindon were on the back foot when Callum Kennedy almost sliced Grant's cross into his own net.
Paynter forced Tomlinson into another near post save, but saw his chances of squaring things up end after he tangled with O'Connor in the middle of the park. Referee Swarbrick spotted the frontman's angry response and sent him down the tunnel for an early bath.
But Swindon's hopes didn't die completely then and Cox came close with two late headers which both crept narrowly past the post.
Crewe caretaker manager Dario Gradi said: "It wasn't a great performance but it wasn't bad considering we had six people missing.
Swindon's caretaker boss Dave Byrne added: "It's the first time since I've been in charge that I have had to say we didn't deserve anything from the game and we've been beaten by the better side."