Rotherham turned in their best first half of the season so far and could have been two up inside the first 10 minutes.
In the sixth minute leading marksman Lewis Grabban fired wide when it was much easier to hit the target and three minutes later Brett Williams missed a great chance, firing into the side netting with only goalkeeper Scott Shearer to beat.
But Williams made up for his earlier miss with 20 minutes gone when he chased Alex Revell's nod on from Conrad Logan's goalkick and showed great composure to lift the ball over Shearer and into the net.
Rotherham came close to a second two minutes before the break when Grabban's drive was finger-tipped over the bar by Shearer.
Crawley by contrast were restricted to just one effort - a wild shot from the edge of the area by Barnett which went high over the bar.
But three minutes after the break Rotherham were made to pay for their misses. The visitors looked a different outfit and Barnett's surging run down the left was finished by a powerful shot past Logan.
And in the 66th minute Crawley's second half improvement brought the winning goal. Substitute Jon-Paul Pittman's pull back from the left was fired against the post by Simpson, but he made no mistake tucking away the re-bound.
Source: DSG