Bogomolov saved match point in the third-set tiebreaker and won on his third match point, handing the sixth-seeded Querrey his earliest exit in eight Memphis appearances.
"Being up match points and not converting, it's easy to lose focus," Bogomolov said. "I just kept believing in myself."
Querrey, the second-highest ranked American in the world behind No. 13 John Isner, was up a break early in the third set but was undone by 33 unforced errors in the final set. Querrey won the 2010 Memphis title.
Sock, who reached the quarterfinals of last year's Indoor, advanced with a 7-5, 1-6, 7-6 (5) victory over France's Adrian Mannarino. Baghdatis, the Australian Open runner-up in 2006, won his first match of the year in three tries by defeating American qualifier Rajeev Ram 6-3, 6-4.
Also advancing was 6-foot-11 Croatian Ivo Karlovic, who tied an ATP World Tour record last week in Zagreb with 44 aces in a first-round win.
Japan's Kei Nishikori, the top seed in the tournament, will play his first match Wednesday. He received a first-round bye. No. 2 seed Feliciano Lopez of Spain and No. 3 Lleyton Hewitt of Australia also got byes and will open Thursday.
Sock beat Mannarino in straight sets last month in Auckland, but struggled throughout this time. Sock served for the match at 5-3 in the third set, but was broken. In the tiebreaker, he took advantage of Mannarino's backhand error to go up 6-5 and won the match when Mannarino softly netted a second-serve return.
Baghdatis, a top-10 player in 2006, won in straight sets.
"Today I played pretty solid and I'm happy I won," he said. "That's all that matters at the end of the day."