Running back Jaylen Warren also suffered an ankle injury and did not return.
Rodgers is in his first season with the Steelers (6-4) after spending the past two years with the New York Jets.
The Steelers got off to a hot start with Rodgers this season, beginning the year 4-1. But Rodgers and the Pittsburgh offense have struggled as of late as the Steelers skidded to a 5-4 mark with last Sunday’s 25-10 loss to the Chargers the low point of the slide.
Rudolph was an efficient 12 of 16 for 127 yards with one TD pass in relief, but the Steelers’ strong second half was thanks in large part to an opportunistic defense that registered two touchdowns (a Kyle Dugger 73-yard INT return and a James Pierre 34-yard fumble return) to pull away from the rival Bengals.