I believe there were fully limited-access grade-separated approaches to the Bay Bridge on the Oakland side (CA-17, US-40, US-50) by 1937 or 1938, the "distribution structure" which was a predecessor to the current MacArthur Maze.
Does that predate the Doyle Drive freeway on US 101?
I believe it does. From my memory of CHPW
articles browsed from 1936 onward (NB: Internet Archive now has the entire run of CHPW
in scanned PDF--almost 2 GB worth of highway goodness), the Distribution Structure opened at the same time as the Bay Bridge proper, but the rail tracks on the lower deck and the Transbay Terminal opened later, around 1937 or 1938. The Golden Gate Bridge opened in 1937 and I believe its accesses on either side opened at about the same time.
* November 1936 is the CHPW
Bay Bridge opening special issue and includes an article on the Distribution Structure, explaining that it was located in Emeryville to allow the Bay Bridge approach roadway to cross two rail lines at once.
* May 1937 is the Golden Gate Bridge opening special issue and has a feature on the construction of what it calls the "Waldo Point road," now known as US 101 just north of the Golden Gate. (This was built by the State of California rather than the Golden Gate Bridge District.)
* May 1938 contains an in-depth account of the Transbay Terminal, whose opening was reported in December 1938.
Edit: I am now going through CHPW
June 1939, which has a feature on the "Presidio Approach to Golden Gate" which was then under construction. It is therefore definite that the Doyle Drive freeway was finished after the Distribution Structure. An interesting question, which I hope later issues of CHPW
will clarify, is whether the Doyle Drive freeway opened before or after the Arroyo Seco Parkway, which (notwithstanding any competing claim the Distribution Structure may be able to make) is usually quoted as California's first length of freeway.