The home side started well and deservedly took the lead on 20 minutes with a close range finish.
The Millers levelled four minutes later when Josh Thornton's long ball was headed into his own net by a Southwell defender after a mix-up with the goalkeeper.
Carlton then took control and took the lead just before half-time when Luke Smithson converted Joe Naylor's through ball.
And the win was sealed in the second half when Wes Henry broke clear to convert Smithson's pass.
A brief match report and photographs will appear tomorrow.
**The Millers play away at Gedling Town on Thursday night before travelling to Sutton Town on Saturday.