The visitors, who resumed on 115 for two in their second innings, 130 runs behind Mumbai's first innings total of 501, batted sensibly to deny the hosts an outright win.
At stumps on day four, Odisha were 356 for five with Natraj and Sujit Lenka (0 not out) on the crease.
Natraj, who hit 130 in the first innings, once again showed good application as he rescued his team by scoring an unbeaten 127 off 218 deliveries, laced with 17 fours and three sixes.
Natraj also put on a solid 147-run stand with Niranjan Behera for the third wicket before the latter was dismissed in the fifth over after lunch.
Niranjan, who too got a reprieve when he was on 28, made a patient 206-ball 59. His knock comprised of nine boundaries, including the one he struck to bring up his half century.
Still nine runs short of equalising Mumbai's total when Niranjan departed, Natraj and captain Biplab Samantray shared a 64-run partnership to take the total to 300.
Abhilash Mallick played a cameo with a quickfire 40, which included seven fours and a six.
From 352 for five, Natraj and wicketkeeper-batsman Lenka kept the rival bowlers at bay for a couple of overs till play was called off after the mandatory overs.
