Striker Broughton also saw red after a second booking, and with an appeal out the question for two yellow cards, both lads will miss this weekend's game against Aldershot.
On reflection, Robins was angry that Broughton's challenge came just seconds after the thought to haul him off crossed his mind.
Speaking to the Star, he said: "We had a conversation with him. But he was doing ever so well up there, causing real problems but, in hindsight, we should have done it - but that's with hindsight.
"I was thinking of taking him off about 25 seconds before he got sent off and should probably have replaced him.
"He's been booked for an innocuous challenge and the referee was threatening to send him off. They probably had that in mind in the second half, probably thought they might get him sent off.
"Drewe twice went for headers and got kicked in the face - the second time their lad has got booked for it."
He continued: "It was always going to be a tough game with Shrewsbury's home record."
"Marc Joseph did the role in front of the back four and did the job asked of him.
"We looked solid and it was the way to go, and with Shrewsbury doing so well here and scoring a lot of goals and hardly conceding any, you have to give them a bit of respect."
The Millers are without a league win in six attempts, picking up just two points since their 2-1 victory at Macclesfield on October 24th.