Jon Russell opened the scoring before Stephen Humphrys came off the bench to double the lead late on.
Joe Rafferty struck a fierce effort goalwards inside three minutes for the Millers, but the onrushing Barnsley defenders were able to block.
The hosts took the lead after 32 minutes through Russell.
After Dawson pulled off a spectacular save to deny his headed effort, he pounced on the loose ball and made no mistake to give his side the advantage.
It looked as though Rotherham were going to level with 18 minutes to play when Jack Holmes was found at the back post, but Barnsley goalkeeper Ben Killip recovered well to keep a close-range effort out.
Humphrys added a second for the Tykes with four minutes remaining when he found himself behind the last line of the Rotherham defence and smashed into the bottom-right corner from outside the box.