Costello was going into a sliding motion against Northwestern when he absorbed a forearm to the head, but the play did not draw a targeting penalty.
"I believe the officials on the field and replay officials missed the call," Shaw said this week. "I thought it was an ejectable offense. I've communicated with a lot of different people in a lot of different places. It was unfortunate it was missed."
Mills, ESPN's No. 2-ranked ranked pocket passer in the Class of 2017, was 7-for-14 for 81 yards in relief of Costello, a second-team All-Pac-12 selection last year.
Shaw also announced earlier this week that All-America-caliber left tackle Walker Little will be out until at least midseason due to a lower body injury suffered against Northwestern.