The 24-year-old Bulgarian, seeded sixth, was outgunned by 2010 Queen's champion Querrey in the first set and fell a break down in the second before rediscovering his touch to level a match which will be completed on Tuesday.
Four-times champion Lleyton Hewitt of Australia lost 6-7 7-5 6-2 to South African Kevin Anderson on his 16th and final appearance at Queen's.
The 34-year-old Hewitt, a former Wimbledon champion, served for the match in the second set, but Anderson, the world number 17, fought back and will face Nick Kyrgios or Stan Wawrinka in the second round.
"This has been one of my favourite tournaments," Hewitt told reporters.
"I love the whole month in London. I'm going to have so many great memories of this tournament."
Third seed Milos Raonic of Canada beat Briton James Ward 5-7 6-3 6-2.
Ward, world number 109, took the first set before Raonic overpowered the home favourite to advance.
British top seed Andy Murray plays qualifier Lu Yen-hsun of Taiwan and Rafa Nadal faces Ukrainian Alexandr Dolgopolov on Tuesday.
(Reporting by Ed Osmond; editing by Justin Palmer)
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