Britt Assombalonga had seen a shot blocked moments before Smith's effort, and the Middlesbrough skipper threatened again shortly after, curling a 20-yard strike just past the right-hand post.
Boro were marginally the more threatening team for the majority of a scrappy first half, but they should have fallen behind when Jonny Howson produced a rare error midway through the opening period.
The veteran midfielder left his back-pass to goalkeeper Bettinelli well short, but after nipping in to claim possession, Crooks side-footed a poor effort past the outside of the upright.
Given that he was clean through on goal, the Rotherham striker should really have scored, but he made up for his error by firing his side into the lead two minutes before the interval.
Smith's shot was blocked, with the ball ballooning into the air inside the Middlesbrough penalty area. George Saville's headed clearance was weak, and after latching on to the ball just outside the 18-yard box, Crooks drilled a crisp low finish past Bettinelli's right hand.
Boro came close to claiming an equaliser at the start of the second half, but while Marc Bola's cross was only cleared to Patrick Roberts, the Manchester City loanee fired over after swivelling in the box.
An even better opportunity went begging shortly after the hour mark, with Duncan Watmore's blocked shot rebounding invitingly for Sam Morsy. The midfielder had time to pick his spot eight yards out but drilled a low shot straight into the legs of Rotherham goalkeeper Jamal Blackman.
It was to be Boro's last realistic chance of getting back into the game, with the Millers doubling their lead with 11 minutes left.
The Middlesbrough defence failed to deal with a free-kick floated into the area, and when centre-half Michael Ihiekwe swooped onto the loose ball on the edge of the six-yard box, he was chopped down by a desperate Bettinelli. Smith stepped up and claimed his fifth goal in the last seven games from the spot.
Rotherham's third goal came in stoppage time and saw 21-year-old substitute Giles stroke home a slick finish after he was teed up by Ben Wiles.
