Wigan Athletic 0 Leeds United 2
Williams, making his first league start after joining from Everton, was yellow-carded on nine minutes for taking out Stuart Dallas in centre-field.
And he received his marching orders for going in late on Bamford just 11 minutes later.
Up to then, Latics had been giving as much as they'd been getting from a side Paul Cook has tipped to win automatic promotion this season.
But from the moment Bamford opened the scoring on 34 minutes, from close range after Adam Forshaw had hit the post, there was only one winner.
Bamford doubled the Leeds advantage midway through the second half when he bundled the ball home from a Barry Douglas corner.
Latics had been fortunate to only concede a corner, after Antonee Robinson had clearly handled a cross from Pablo Hernandez.
That was about all they got all afternoon from referee Andy Madley, who gave Cook a yellow card 13 minutes from time for dissent.
Wigan's best moments came from a couple of brilliant long-range free-kicks from Lee Evans, but Leeds goalkeeper Kiko Casilla was equal to them.