The Ivorian was shown a red card after headbutting Ezgjan Alioski off the ball early in the second half, leaving the ten-man Gunners to dig in for a point, as their run without a goal from open play extended to five Premier League games.
Leno did well again to keep out a fierce drive from Stuart Dallas and then substitute Rodrigo crashed a shot against the bar from the edge of the area.
Arsenal nearly grabbed a winner, against the run of play, when Hector Bellerin put Bukayo Saka through on goal but he was unable to round Leeds keeper Illan Meslier.
Leeds struck the woodwork twice in the latter stages with Bamford heading against the post and Raphinha also striking the upright deep in stoppage time.
Arsenal are 11th on 13 points from nine games after their first draw of the season with Leeds 14th on 11.
Additional reporting by Reuters.