Londoner Ward eased past Serbian Boris Pashanski 6-4 6-3, while Scotsman Baker defeated Educador’s Julio Cesar Campozano 6-3 7-5.
Ward, ranked 247th in the world, and the 246th-ranked Baker are now two more wins away from the men’s singles main draw via the three-round preliminary process.
"I started out pretty well but then I got a little bit tight into the middle of the second set," Baker told BBC Sport.
"I've known Campozano since he has a junior; he always fights hard and it became a bit of a dogfight, but it's good to get through.”
Olympic gold medallist Andy Murray, seeded third for the tournament, took a courtside seat for good friend Baker’s win.
Baker said: "Andy popped in. He actually checked to see if I was alright about him coming out - he could be a bit of a tough critic!"
Qualification for the women’s singles draw begins on Thursday, with Johanna Konta and Anne Keothavong representing Great Britain.
Konta has been paired with Latvian Anastaija Sevastova, and 22nd seed Keothavong will meet America’s Grace Min.