The Scot joined Championship side Blackburn Rovers on a season-long loan on deadline day, leaving the Whites without a specialist senior left-back.
“It makes me very sad that he has left our team because during the two years he has been with us he has made a significant contribution and he's an impeccable professional and a very respected member within the team," said Bielsa.
“At left-back I opted for Dallas, (Ezgjan) Alioski and (Leif) Davis ahead of Douglas.
“They're decisions I have to make. For me they have been very difficult. Douglas deserved to play for Leeds and he gave his all to achieve this. I wasn't able to help him enough for him to consolidate himself in our team.
“He's a player who has given huge effort and I regret a little bit that he wasn't able to triumph with us.
“Finally it was decided between myself, the club and Douglas that he should leave for Blackburn. I wish him all the best because he really deserves it. He is a player who is full of virtues and has done some very important things for us."
The Argentine also recalled: “I can't forget that he played a game for us where he had a knee injury for half an hour.
“That's not very common. He thought about the team before himself and he put at risk his knee, which is a very complicated place for a player to get injured."