Rotherham striker Adam Le Fondre fired over the bar inside the opening minute but it was the visitors who made all the early running at the Don Valley Stadium.
Millers goalkeeper Andy Warrington was on hand to keep out Brett Pitman and Lee Bradbury, while Rotherham had a penalty claim turned down at the other end when Rhoys Wiggins appeared to handle the ball on the line.
Rotherham opened the scoring after 26 minutes when Tom Pope headed home Marc Joseph's cross, but the visitors levelled within two minutes of the restart when Pitman skipped round Pablo Mills and raced into the area to beat Warrington with a clinical finish.
Pitman turned provider to help put the Cherries in front shortly before the hour-mark when his cross was headed out by Rotherham and Hollands was on hand to drill the loose ball home.
Bournemouth sealed the win when Pitman's corner was headed back across goal by Steve Fletcher for Hollands to tap in from close range with 23 minutes remaining.