Led by Awana, Delhi bowled out Odisha for 379, 63 runs behind their first innings total of 442.
Delhi were 49 for two at the end of day's play with skipper Gautam Gambhir back in the hut for just five. The other batsmen to get out was Unmukt Chand (3).
India discard Virender Sehwag remained unbeaten on 32 as he smashed seven fours in 31 deliveries, alongside Mithun Manhas (4).
Natraj recorded 157-run stand with Abhilash Mallick (71) for the fifth wicket and looked good to surpass Delhi's first innings total but both fell in quick succession to hand an all important lead to the visitors.
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Natraj scored 23 boundaries in his 290-ball knock while Mallick took 105 deliveries for his 71 that included 10 fours and two sixes.
Besides Awana, left-arm spinner Varun Sood (3/67) was instrumental in removing the lower-order batsmen while Ashish Nehra and Sumit Narwal claimed one wicket each.
Brief Scores:
Delhi - 442 and 49/2 in 12 overs (Virender Sehwag 32 not out; Suryakant Pradhan 1/16)
Odisha - 379 all out in 115.3 overs (Natraj Behera 158, Abhilash Mallick 71; Parvinder Awana 5/97).