Bhandari and Gill scalped three wickets each to shut out Railways for 201 on the penultimate day of the four-day match at the Jamia Millia Islamia university ground here.
At close, the hosts had consolidated the advantage and were placed at 101 for one with Shikhar Dhawan on 55 and Rajat Bhatia on 14 at the crease.
Test opener Akash Chopra was the batsman to be dismissed for a well paced 31.
Resuming at 147 for five and needing 70 more to overhaul Delhi's total of 216, Railways were setback when they lost the experienced Yere Goud in the first over of the day to Gill.
Gill, who picked eight wickets in the previous match against Andhra, then removed dangerous Harvinder Singh (9) when he had him caught behind for final figures of 3-48.
In between Bhandari had leg before Andhra import Shankara Rao (5) before rapping wicketkeeper batsman Sudhir Wankhede (4) to reduce the 'visitors' to 181 for 9.
Siddharth Verma, who had resumed at his overnight 16 and watched wickets fall at the other end, then added 20 runs for the last wicket with Kulamani Parida (7) before the latter was run out.
Verma remained unbeaten on 44 from 138 balls after a near four-hour stay at the crease.