Substitute goalkeeper Phurba Lachenpa did the star turn in the penalty shootout for the Islanders.
FC Goa squandered the lead twice in the 2-2 draw.
Hyderabad FC failed to qualify for the play-offs after being held by the Gaurs.
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.
The draw, Goa's sixth on the trot, helped the Gaurs climb to the third spot on the table.
Mumbai took the lead through Hugo Boumous (20th minute) and Adam Le Fondre (26th minute) but goals from Glan Martins (45th minute) and Igor Angulo (51st minute) made it all square for FC Goa.
NEUFC eventually grew into the game and fashioned better chances in the second quarter of the opening half. They had a penalty claim denied after the half-hour mark.
The Gaurs went ahead through Igor Angulo's 39th-minute goal but his strike was neutralised by SCEB skipper Daniel Fox (65') who scored his first ISL goal.
Goa remain in third spot with 20 points from 13 matches, while Blasters moved up to seventh in the 11-team competition.
Kibu Vicuna's men have grabbed seven points from their last three games as compared to six in their first nine matches.