Divij and Bambridge will now play the second seeded Australian pair of Max Purcell and Luke Saville at the ATP 250 tournament.
In other second round matches played on Wednesday, 22-year-old American Frances Tiafoe took 1 hour 20 minutes to knock out second seed Miomir Kecmanovic of Serbia 7-5, 6-3 in a battle of upcoming stars.
Seventh seeded American Tommy Paul and Belarusian Egor Gerasimov also moved into the last eight of the inaugural ATP event in Kazakhstan.
Tiafoe's American compatriot seventh seeded Tommy Paul outplayed Radu Albot of Moldova 6-4, 6-0 in 1 hour and 12 minutes while Belarusian Egor Gerasimov battled past Czech player Jiri Vesely 4-6, 6-3, 6-4 in 1 hour 50 minutes to make it to the quarterfinals.
Australian fourth seed John Millman played solidly to tame crafty left handed and former world no 7 Fernando Verdasco of Spain to win 6-3, 6-4.
Mens Singles (Second Round)
[7] Tommy Paul (USA) d Radu Albot (MDA) 6-4, 6-0
[4] John. Millman (AUS) d Fernando Verdasco (ESP) 6-3, 6-4
Egor Gerasimov (BLR) d Jiri Vesely (CZE) 4-6, 6-3, 6-4
Frances Tiafoe (USA) d [2] Miomir Kecmanovic (SRB) 7-5, 6-3
Doubles (First Round)
Luke Bambridge (GBR)/Divij Sharan (IND) d Ariel Behar (URU)/Gonzalo Escobar (ECU) 7-5, 4-6, 10-6
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