The visitors took the lead with just 10 minutes gone when new signing Nick Ward headed home a Ricky Bridge corner.
They made it 2-0 midway through the half after a great passing move between Danny Smith and Tim Sanders put Aiden Matranga through on goal.
His shot came back off the post, but last season's top scorer Dave Walker was on hand to tap home.
Another strike from Ward 10 minutes later effectively ended the game as a contest when he converted another Bridge corner.
Kidsgrove had to wait until 16 minutes from time for their fourth when Liam Shotton, who had caused the Stamford defence all sorts of problems, fired in from 20 yards.
It could have been more in the closing stages, but sub Jamie Pointon was denied by two excellent saves from the home keeper.
"We played really well and had too much for them in every department," said manager Peter Ward.
"I'm delighted to have a quality player like Nick Ward come in after we lost Tom Schwarz – he has experience and showed he can score goals."
Boss Ward was also full of praise for Shotton, who spent a lot of last season on the bench because of the partnership forged by Dave Walker and the now-departed Matt Haddrell.
"Liam had to be patient last year and he understood that the lads in the side were both scoring goals," added Ward.
"But he and Dave (Walker) have played together a lot in pre-season and Liam has improved his overall game."