United wasted several chances to secure victory at Elland Road and conceded late on to drop points for the third time in four home league games.
Derby, who beat the Yorkshire club in last season's Championship play-offs, had been outclassed for long periods before substitute Martin rounded off a fine team move with a cool low finish past Kiko Casilla.
Max Lowe's unfortunate own goal had looked likely to be the difference between the two teams as the Rams posed little threat.
"This has happened a lot of times to us, but today was the worst," said Bielsa. "They shot once all match. We created maybe 10 or 12 chances.
"We played 60 minutes very good, maybe the best since I've coached the team. But the last 15 minutes was flat.
"We couldn't control the end of the match.
"When we lose a point in this way it's difficult to admit reasons why. But we started to play long rather than line to line."