Bob Odenkirk was hospitalized for a "heart-related incident" ... sources with direct knowledge tell TMZ, and it was touch-and-go for the last day but it looks like he's now out of the woods.

Bob is awake inside an Albuquerque hospital and we're told he's on the rebound, but when he was rushed to the hospital Tuesday after collapsing on set it did not look good. We're told he was not conscious during the early hours in the hospital, although he's now lucid.

Bob's rep tells TMZ, "Bob and his family would like to express gratitude for the incredible doctors and nurses looking after him, as well as his cast, crew and producers who have stayed by his side."

The rep goes on ... "The Odenkirks would also like to thank everyone for the outpouring of well wishes and ask for their privacy at this time as Bob works on his recovery."

TMZ broke the story ... Bob was rushed to a New Mexico hospital Tuesday night after collapsing on the set of his show "Better Call Saul."

Production sources told us Bob and the rest of the cast were shooting when the incident happened, with Odenkirk going down and being immediately surrounded by crew members as they called an ambulance.