Bamford’s 64th minute 20-yard free kick deflected off the defensive wall to beat keeper Steve Smith.
Despite having debutant striker Danny Fletcher sent off for retaliation in the 16th minute, the 10 men of Rainworth were too good for a Clipstone side whose winless run extended to 30 games.
Only wayward finishing, desperate defending, and excellent goalkeeping by Smith prevented Rainworth from winning by a bigger margin.
The home goal led a charmed life in the first half, with Ethan Wiestort’s lob tipped on to the bar.
Former Wren Rob Paling almost punished his old club’s wastefulness in the opening seconds after the interval with a rising shot from distance that rattled the bar.
But that was as close as the home side came. At the other end Wiesztort was twice denied before Bamford’s goal sealed the points.