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.
Blasters failed to play as a cohesive unit and looked like a group of individuals with no plan.
Despite the win, Odisha stay at the bottom of the table with five points from nine games, while Blasters are in 10th spot with six points from nine matches in the 11-team league.
Odisha lie bottom of the table, and are yet to win a game or to keep a clean sheet and are probably Blasters' best chance to get three points in a packed schedule.
NorthEast have 11 points from eight games and are in fourth spot, while Odisha occupy the tenth place with just two points from seven matches.
Ashique suffered the injury in the second half after his face was caught by the knee of Odisha FC's Jerry Mawihmingthanga.
This was Odisha's fifth defeat of the season and they are yet to record a win.
FC Goa edged Odisha FC 1-0 to climb to fourth spot.
Odisha wrestled control of the match after the first 20 minutes in which Mohun Bagan dominated possession.