Carlos Delgado (27') and Xisco Hernandez (41') gave Odisha a first-half lead before Bobo (65') scored from the spot to set up a nervy finish. Odisha were awarded a penalty at the other end, which Martin Guedes converted to secure three points. Rohit Kumar (89') pulled one back at the death for the home side.
Odisha had Vinit Rai sent off for a second bookable offence in the 65th minute, but they soldiered on for the rest of the match.
After eight matches, Odisha are sixth on the table with nine points while Hyderabad are at the bottom with four points.
The hosts' intent to get off the mark quickly was evident as they pressed high up the field in a bid to launch one attack after another. Aridane Santana flicked a good cross-field ball by Diawandou Diagne into the path of Nandhakumar Sekar who was dispossessed by Adil Khan with a brilliant tackle just before the winger could get a shot away.
The intensity bore fruit as Xisco, who was ever-present in midfield, set up the first goal with a dangerous in-swinging delivery from a corner. Goalkeeper Kamaljit failed to punch the ball away and Delgado made full use of the chance to head home the opening goal.
A curling effort from Robin Singh flew inches wide as Hyderabad attempted to equalise but it was the home team who struck again before the break. The creator turned scorer as Xisco initiated a threatening move from the centre and passed it to Jerry Mawihmingthanga who returned the favour with a pin-point cross that the Spaniard slotted home from inside the box.
Odisha were jolted when Vinit Rai handled the ball inside the box, conceding a penalty, and was sent off for picking up a second yellow. Bobo made no mistake from 12 yards to give his team a chance to get back into the game.
Hyderabad's hopes of a comeback were soon ended as Giles Barnes fouled substitute Martin Guedes and conceded a spot-kick at the other end. Argentine Martin Guedes stepped up and coolly slotted into the net to take the game away from Phil Brown's men.
Hyderabad reduced the deficit in the 89th minute when a hopeful cross into the box by Gurtej Singh was headed in by Rohit Kumar but it was nothing more than a consolation goal.
( With inputs from IANS )