In his third and final qualifying match, the 25-year-old player struggled with his serve in the opening set before Polanksy pocketed the set with ease.
But he bounced back in the remaining two sets and won the match 1-6, 6-3, 6-3 in one hour and 55 minutes.
The final set followed a similar pattern and Bhambri grabbed the only break point that came his way to book his berth in the main draw of this year's first grand slam.
World No. 139 Ramkumar came back to win the second set but eventually lost 4-6, 6-4, 4-6 in one hour, 44 minutes.
(This story has not been edited by Business Standard staff and is auto-generated from a syndicated feed.)