Watched by his father, a renowned tennis coach in the former Soviet Union, the American made a nervous start by dropping the first set and then gave up a 4-1 lead in the deciding set before winning the match.
"Four-four (in the third set) was serious, but I returned well a couple of times and then finished fine and that's the main thing," Bogomolov Jr. said.
Bogomolov Jr. was born in Moscow but moved to Mexico with his family when he was 9 and then to the United States about two years later. His father coached Yevgeny Kafelnikov and Andrei Medvedev, among others.
Bogomolov Jr. is making only his second visit to Moscow, where his father lives, in the past nine years. He's accompanied by his Brazilian wife and their 2-year-old son. He said his young son is so confused by the three languages in his life that he has hardly said a word.
Kuznetsova beat French qualifier Alize Cornet 6-3, 6-3, while Muller had 19 aces on his way to a 7-6 (5), 6-4 victory over Russian qualifier Konstantin Kravchukin. Tursunov defeated Spain's Albert Ramos 7-6 (5), 6-4.
The 19th-ranked Kuznetsova broke Cornet twice to lead 4-1 in the first set. The two-time Grand Slam winner rallied from 2-0 down in the second set, winning five consecutive games to take it in front of the home crowd.
The 50th-ranked Muller saved two break points in the opening game and prevailed in a first-set tiebreaker before scoring a decisive break in the first game of the second set.
In women's first-round play, Tsvetana Pironkova advanced after Petra Martic retired with a left foot injury after losing the first set 6-1. Ksenia Pervak was another Russian to reach the second round after a 7-5, 6-4 victory over Galina Voskoboeva of Kazakhstan.
Pervak will play third-seeded Marion Bartoli, fresh off a Japan Open win.
"She is in perfect form now, but I'll try my best to win," Pervak said.