Delhi now need 174 more runs on the final day tomorrow with six second innings wickets in hand to keep their qualification hopes from Group B alive.
Openers Dhawan (16) and captain Gambhir (34) fell early in the Delhi second innings -- inside 15 overs -- before their side had two more cheap dismissals, that of Unmukt Chand (11) and Nitish Rana (10) in the space of three deliveries, to find themselves staring at a defeat.
At stumps, Delhi were 109 for 4 from 30 overs.
A loss for Delhi in this match would knock them out and Odisha will join Karnataka and Jharkhand as the three qualifiers for the knock-out stage from the group.
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Delhi were to blame themselves for this precarious situation as they had pinned Saurashtra on the ropes yesterday by sending back five batsmen for 130 runs on board in their second innings.
The eighth-wicket pair of Vasavada, who resumed the day on 25, and Unadkat stitched 135 runs and took Saurashtra from 261 for 7 to 396 for eight before Pradeep Sangwan broke through.
Saurashtra's tail did not last long as they were soon bowled out for 420.
For Delhi, Navdeep Saini took three wickets for 122 runs while Nitish Rana got two wickets. Sumit Narwal, Pradeep Sangwan, Pulkit Narang and Manan Sharma picked up one wicket apiece.
Saurashtra: 92 & 420 all out in 127.4 overs (Kishan Parmar 149, Arpit Vasavada 89, Jaydev Unadkat 80; Navdeep Saini 3/122).
Delhi: 237 & 109/4 in 30 overs (Gautam Gambhir 34; Shaurya Sanandia 3/39).