Baker fought back from 4-2 down in the deciding set at Roehampton to defeat Italian 17th seed Matteo Viola 4-6 6-3 6-4, while Bogdanovic was a 7-6 (7/5) 6-3 winner over Slovakia's Marek Semjan.
Baker will hope there is the same outcome as the last time he beat Viola, when he went on to qualify for the main draw at the Australian Open in January.
The Scot's five appearances at Wimbledon have all come through wild cards, most recently last year when he lost in the first round to Andy Roddick.
Bogdanovic has not played at Wimbledon since receiving the last of his eight wild cards in 2009, all of which ended in first-round defeats.
But the 29-year-old has been in good form this year and at 274 in the world rankings is the British number five.
There was a first win in grand slam qualifying for 24-year-old Rice, who defeated Germany's Sebastian Rieschick 6-3 6-2 to become the third British man into round two.
He had won the LTA's wild-card play-off to earn a spot in the qualifying draw.
There were disappointments too, in particular for Dan Evans, who could not build on his fine grasscourt form this season and went out 6-1 3-6 6-2 to Spain's Daniel Munoz-De La Nava.
Evans has built on his starring role in Britain's Davis Cup win over Russia in May with several fine wins, including last week at Queen's Club, where he beat two top-100 players.
That had led to speculation the talented 23-year-old might be given a wild card into the main draw of Wimbledon but instead he was given a place in qualifying.
He fought back well after a one-sided first set but made a poor start to the decider and did not recover.
Josh Goodall, Joshua Ward-Hibbert, Lewis Burton and Ed Corrie also lost, although the latter gave ninth seed Illya Marchenko a good fight before going down 6-4 5-7 7-5.
In the second round, Rice takes on Flavio Cipolla, Bogdanovic plays Evgeny Korolev and Baker meets Igor Kunitsyn, with three wins needed for a place in the main draw, which gets under way on June 24.
The women's qualifying draw, which features eight Britons, starts on Tuesday.