Braces from Danny Gordon and Aaron Hooton plus goals from William Mellors-Blair and Jordan Wilson saw the Millers complete a comfortable victory over the Badgers who play their football three tiers below Carlton.
It might have been different had Eastwood taken one of a couple of early opportunities, but once Gordon fired home a free kick on 20 minutes the result was never in doubt.
Carlton played some fine football and were too sharp for their opponents as they moved into the quarter-finals.
A full report will be on site tomorrow. Carlton are next in action on Saturday at home to Chasetown in the league.