The win was deserved for a Carlton side who always had the edge on their opponents and, but for some wayward finishing, the margin of victory would have been greater.
Manager Wayne Scott made four changes from the side which lost against Frickley last time out. In came Oliver Robinson, Kieran Walker, James Frost and Louis Tomlinson in place of the suspended Riece Bertram, the absent Ben Hutchinson, the injured Khyle Sargent and Niall Davie.
Lincoln featured ex-Miller Rob Norris amongst their substitutes.
The Whites started the better with Jack Steggles fumbling an early cross over the line for a corner, which led to a clear chance for Matt Cotton but his effort flew just wide.
Luke Smithson, once again impressive in a wide right position, lifted a lovely chip over Seth Greeves but just over the bar.
Greeves then completely missed a long throw by Walker but Tomlinson was unable to get a head on the ball at the back post with the goal gaping.
Minutes later Greeves made a comfortable save from a Walker header after a Connor Bartle free kick, before Eddie Clarke headed just wide from a Robinson corner.
Another long throw from Walker again found Tomlinson at the back post but he failed to connect properly with the ball and the chance was saved.
By now Carlton should have been well ahead so it was no surprise when it was Lincoln who struck first. The Millers tried to play an offside trap but Cotton ran onto a through ball, rounded Steggles and tapped home.
Lincoln’s tails were up and they spent the next five minutes camped in the Carlton half before the Millers equalised with a stunning goal. Robinson played a free kick to the edge of the Whites box. Opoku chested the ball, held off his marker, swivelled and fired low into the corner of the net in front of the travelling support.
Forward came the Millers and Walker fired just wide shortly before half time.
Carlton continued to put the home side under pressure and took the lead 10 minutes into the second half. Smithson went on another excellent run to earn a corner. When Bartle’s delivery was played back to him he sent over an inviting cross to the back post which Oliver Clark headed powerfully past Greeves.
Lincoln had a rare attack but Steggles made a comfortable save from a Tom Garrick drive. Carlton were bossing the midfield with Oliver Clark particularly impressive and Opoku running the defence ragged.
Eddie Clarke headed over from another Bartle corner while Niall Davie twice just failed to get to ball at the back post sent over by Smithson.
As it was, Carlton saw out the game without any difficulties to earn their place in the next round.
Carlton Town: Steggles, Robinson, Bartle, Moore, E Clarke, Walker, Frost, O Clark, Opoku (Kwadu 72), Smithson, Tomlinson (Davie 46). Unused substitutes: Ilesanmi, Pick, Stokes
Goals: Opoku 33, O Clark 55; Matt Cotton 27
Carlton Town Supporters Club MOTM: Luke Smithson