Jack Harrison whipped in a superb cross from the left wing and Bamford rose well to score for the third straight game and leave Chris Wilder's side without a point or a goal this season.
The game had been a story of excellent goalkeeping at both ends.
Meslier was alert again after the break when George Baldock turned sharply on the edge of the box to lose his marker and unleash a powerful drive but the 20-year-old keeper parried well.
Helder Costa then threatened with a stinging volley that was kept out by the diving Ramsdale.
Bamford's fine header, however, ensured that Leeds move to six points from their first three games on their return to the top flight after a 16-year absence.
"It was a just result," Bielsa said. "The first half was even. We could've scored but also conceded. The second half was a lot better for us. I think we created a small but just difference to create the victory."
"We'll be glad to see the back of September. Fresh start in October at Arsenal away. I can't fault a lot of things the players did in the game. I thought it was really tight, you have got to take the chances in the Premier League.
"We have to be more clinical, the best chances fell to us, we know how Leeds play, high energy and high tempo. We ultimately got punished by a fantastic ball and a great header."