A brace from two Indian youngsters Paul Ramfangzauva (49', 66') and Jerry Mawihmingthanga (51', 67') enabled Odisha to end the season on a high, with S Lalhrezuala (33') and Diego Mauricio (69') chipping in.
A win for Mumbai City FC in their final game against ATK Mohun Bagan would seal them a spot in next season's AFC Champions League group stage.
Goals from Alberto Noguera (18th min) and Jorge Ortiz (26th min) gave Goa an early lead to the Gaurs before Diego Mauricio (30th min) pulled one back for Odisha.
Expecting a team that has conceded 29 goals in 17 matches to do well in a highly competitive tournament like ISL is pointless.
The draw left Blasters in ninth spot with 16 points from 17 matches, while Odisha stay rooted at bottom with just nine points from 16 games.
The result meant the Mariners closed the gap with Mumbai to three points, while Odisha's top-four chances are all but over.
The club announced that an interim coach for the remainder of the season will be named shortly.
Jamshedpur moved up to sixth spot with 18 points from 15 matches, while Odisha stay in bottom place with just eight points from 14 games.
Halicharan Narzary put Hyderabad ahead at the break but saw his header cancelled out by Cole Alexander in the second half.
Goncalves scored in the 15th and 21st minutes for Chennayin before Diego Mauricio struck a screamer in the 64h minute to give Odisha some hope.