Karnataka were staring at a defeat when they started the day at a precarious 244 for six, just 81 ahead in their second innings, but eventually put on board 393.
Thanks to fighting knocks from Gautam, Gopal and Gowtham (46), they set Odisha a target of 231.
Odisha ended at 63 for no loss and walked away with three points thanks to their first innings lead.
After Gautam was dismissed five runs short of his 11th first class hundred, Gowtham joined hands with Gopal and the two raised a crucial and match-saving 86-run partnership for the eighth wicket.
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Odisha's left-arm spinner Dhiraj Singh (5/98) completed his five wicket haul while Govinda Poddar took two wickets.
Brief Scores
Karnataka 179 and 393 in 121.2 overs. (C Gautam 95, R Samarath 49, S Gopal 77, K Gowtham 46; D Singh 5/98).
Odisha 342 and 63 without loss in 32.Overs.