IMVHO, Miller won the first round. Bisping started landing shot after shot and, TBH, both fighters looked a little gassed. Miller, actually gave up. When he went to his corner, he didn't even look dazed. His face was fucked up, but, he looked fully coherent and just said "I'm really tired."
I honestly expected more, I didn't think Mayhem would go 5 full rounds, but, I certainly didn't expect what I saw out of him. It's like he gassed himself keeping Bisping on the ground in that hybrid leg/body triangle he had going.