The Millers started the better and Sammy Ile opened the scoring on 10 minutes, latching onto a poor back pass and slotting home after taking the ball around the goalkeeper.
Basford improved and Harry Wakefield equalised on 29 minutes, heading home a cross from the right, before Kieran Fenton put the visitors in front with an excellent free kick into the top corner from 20 yards.
Riece Betram put through his own goal on 73 minutes to extend Basford's lead before Luke Smithson, back at Stoke Lane, reduced the deficit four minutes later with a well taken finish from inside the area.
Both sides had further chances before the game ended on another red-hot day.
Carlton are back in action next Saturday at home to Bedworth Utd.