The visitors took the lead just before the break when City goalkeeper Dan Bentley unwisely left his line to meet a Wes Harding cross from the right and was beaten to the ball by Smith, who headed home.
Skipper Wood doubled Rotherham's advantage with a close range header from Dan Barlaser's 74th minute corner as City slumped to a sixth successive home defeat.
Three of those have come since Nigel Pearson's arrival as manager and he cut a frustrated figure in the dugout as his side were outbattled.
Pearson made three changes from the goalless midweek draw at Blackburn, bringing in fit-again goalkeeper Bentley and midfielders Callum O'Dowda and Liam Walsh, both sidelined until recently with long-term injuries.
With manager Paul Warne, still in self-isolation, Rotherham reacted to the 4-1 home defeat by Watford with four changes, goalkeeper Viktor Johansson coming in alongside Lewis Wing, Wes Harding and Ryan Giles.
Already on a club record run of home defeats, City began brightly and Walsh's early volley from just outside the box was too high.
The Millers threatened on 13 minutes when Michael Ihiekwe's low ball into the box was turned towards goal by Smith but the effort was weak and did not trouble Bentley.
Nahki Wells was wide with a free-kick for the home side midway through the first half as, with Walsh showing up well, they pushed forward with menace.
Neither goalkeeper had been called upon for a serious save when Smith broke the deadlock on 44 minutes, leaving Pearson shaking his head in the dugout.
Rotherham had a chance to increase their lead early in the second half, but Freddie Ladapo's cushioned volley was too close to Bentley.
Three minutes later Wells had a volley from a narrow angle tipped wide by Johansson.
Bentley looked to have kept City in the game with a flying save from Ihiekwe's 73rd minute header but it was the resulting corner that led to Wood making it 2-0.
Pearson introduced youngsters Sam Bell and Sam Pearson in the closing stages, but although Johansson made a couple of late saves, Rotherham had a firm grip on the points.