Rodrigo - who returned from injury in Monday's game at West Ham - came off 11 minutes from time when he was replaced by Mateusz Klich. It is the second time this season that the 29-year-old has been replaced after coming on as a substitute - with the only occasion coming when he was replaced in the game at Sheffield United in September.
Bielsa, whose side drew for the first time in 19 matches - avoiding a club record run without a draw - and turned in a good team performance against a side who are unbeaten under Thomas Tuchel, said: "He (Rodrigo) has been without playing for a long time and it is natural that it cost him a little bit in adapting to the rhythm.
"I value a lot the effort he put in when he was on the field because it is an effort that is above what he is actually capable of at this moment in time.
"The contribution of Rodrigo was above what he is actually capable of doing at this moment. More than apologising to him, I have to thank him for his contribution on the pitch."
On his side's display, he added: "We value a lot the point, if there had to be a winner it wouldn't have been us. In the first half they established a difference in comparison to us.
"In the second half we were slightly better. The superiority they showed in the first half was greater than the superiority we showed in the second half. If we don't play at the rhythm we play at, the differences between us would be a lot bigger."
Meanwhile, Bielsa confirmed that captain Liam Cooper missed the game through sickness, with Pascal Struijk replacing him in the line-up, while he had no details regarding Bamford's injury.
Offering his take on the game, Blues chief Tuchel said: "We had enough chances to score. It was an OK game, sometimes it is hard to score. It is not that you get what you wish for and we have to keep on working hard and be more precise in the opponents' box.
"It was not so nice to play and the pitch is not in good condition - this is a factor. Our timing in the last 20 minutes - we need to be more clinical. We invest a lot against the ball so also we were tired when we arrived in good positions."