Diatchenko, who resides in the Russian capital, fought back from a set down in the final match of the day to prevail 4-6 6-2 6-2 in a contest that lasted just short of two hours.
Ekaterina Makarova and Flavia Pennetta, the second and third seeds respectively, crashed out on Wednesday and world number 11 Cibulkova followed them with a shock second-round exit.
The Slovakian looked to be on course for the quarter-finals when, after breaking in the first game of the match, she claimed the first set despite failing to hold serve on two occasions.
Diatchenko must have feared the worst when she was broken again in the opening game of the second set, but she hit back immediately with a break of her own and then won a marathon fourth game to lead 3-1.
A further break ensured she levelled the match and the 24-year-old picked up where she left off in the decider, breaking four times as Cibulkova's service woes continued to claim a major scalp.
Diatchenko can now look forward to an all-Russian quarter-final against Anastasia Pavlyuchenkova, who beat Alison Riske 6-2 6-2 in the second round.
Moscow-based Pavlyuchenkova needed just 59 minutes to get the better of American Riske, raising hopes of a tournament victory on home soil.
World number 30 Pavlyuchenkova sent down nine aces and won half of the 12 break points she fashioned as Riske, 14 places below her opponent in the WTA rankings, toiled.
Tsvetana Pironkova of Bulgaria was also a second-round winner on Thursday at the expense of Serbian wildcard Aleksandra Krunic, while qualifier Katerina Siniakova beat France's Kristina Mladenovic in straight sets.