Fish and Baker boosted their nation’s hopes, which were dampened by the opening day exits of five-time champion Venus Williams and big-serving John Isner, AFP reported.
Fish made a winning return to tennis in his first match following a procedure to address a frightening heart scare.
The world number 12 beat Spain’s Ruben Ramirez-Hidalgo 7-6 (7/3), 7-5 7-6 (7/1) and looked comfortable moving around the court.
Meanwhile, Baker, who lost six years of his career to an assortment of injuries and underwent five operations, clinched a first ever win at Wimbledon, beating Portugal’s Rui Machado 7-6 (7/2), 6-4, 6-0.
The 27-year-old reached the second round of the French Open last month in what was his first Grand Slam appearance since the 2005 US Open.
He will next tackle Jarkko Nieminen of Finland for a place in the last 32.