Manager Graham Drury made two changes from the win over Stocksbridge Park Steels with Jordon Cooke coming in for the unavailable AJ Adelekan and Harry Limb replacing Jack Duffy.
Carlton has always been a bogey side for the Daniels with especially at the Bill Stokeld stadium.
The Daniels started brightly nothing to test Jack Steggles in the home goal.
With about 15 minutes gone, Stamford had a strong penalty shout turned down by the referee when Harry Limb was tackled in the box.
The was compounded when Carlton took the lead on 21 minutes when a cross from the left from Niall Davie found substitute Joel Daft unmarked at the far post to fire home.
Stamford continued to press the home side and put the Carlton goal in almost constant pressure and captain Sean Wright has a shot saved before Tom Siddons twice had efforts saved by Steggles.
Five minutes before half time, Stamford were forced into a change with Harry Limb going off injured with James Hicks replacing him.
At half time, manager Graham Drury was forced into another change with livewire Cosmos Matwasa going off to be replaced by the evergreen Jon Challinor.
Carlton had a couple of free kicks early in the second half but nothing to trouble Dan Haystead.
Stamford continued to press Carlton and forced a series of corners which the home team managed to repel.
It was beginning to look like the Daniels were going to come away with nothing as time ran on and chances squandered. Mike Armstrong had an opportunity as he strode forward from defence and he let fly from 25 yards which was just wide of the goal.
The Daniels pressure finally paid off with 13 minutes remaining when a Harry Vince corner found Wright at the back post to head home from close range.
Stamford nearly claimed victory in the closing minutes when James Hicks held off the Carlton defenders to flick the ball goalwards only to see it comes back off the crossbar.