The win was fully deserved after a thrilling performance from the Millers, who created chance after chance and were denied by a combination of the woodwork and some inspired Sutton defending before Opoku finally broke the deadlock with 13 minutes remaining.
Opoku himself hit the post in the second minute before seeing Lewis Gwilliams save excellently from him shortly afterwards.
Nat Watson beat the keeper with a fierce effort only to see Alex Smith volley the ball off the goal line before Niall Davie hit a stunning free kick that crashed against the cross bar - all inside the first 15 minutes.
Davie and Opoku hit efforts inches wide, Toby Moore saw a shot deflected over the bar and the Sutton defence and goalkeeper made numerous blocks to keep their goal intact in a game which the Millers in general dominated.
The visitors did create a couple of decent chances which forced Kieran Preston into making some fine saves, but in general the Carlton defence had the measure of Sutton and it was only really a question of whether the Millers could finally breach the Sutton defence.
The goal finally came on 77 minutes when Opoku powered home a header after a corner wasn't cleared. Czerwak then sealed the win seven minutes later when he fired home a Davie cross having started the move himself deep in his own half.
The win, a fourth on the spin, sees Carlton move up to third place in the table with 13 points from five games. They return to action on Saturday with a trip to Yaxley.
A full report from tonight's game will be on site tomorrow.