The former Chelsea manager came under fire after two draws against unfancied rivals which cut Partizan's lead at the top to six points as neighbours Red Star, looking for their first league title since 2007, had closed in with a pair of wins.
"We played better football on Saturday [in a 0-0 home draw with Sloboda Sevojno] but we are delighted to have won today because three points from this game is all that matters," Grant told a news conference.
"It is a nice feeling to get off the mark, we deserved this victory and should have scored more goals."
However, Partizan were fortunate to come away with the spoils on a poor and rock-hard pitch after the home side had a strong penalty claim waved away before visiting goalkeeper Vladimir Stojkovic also denied them with two excellent saves.
Midfielder Nemanja Tomic fired Partizan ahead with a superb free kick from 30 metres and Bulgaria defender Ivan Ivanov headed the winner after Ecuadorean striker Augusto Batioja had equalised with an unstoppable header.
Partizan's teenage forward Lazar Markovic then got away with hauling down Batioja on the stroke of half-time and Partizan hung on in the second half as Stojkovic came to their rescue.
Red Star kept up the pressure with a 4-0 home win over Smederevo as their vociferous fans again turned up in numbers at the Marakana stadium.
The 1991 European Cup winners delighted 40,000 faithful supporters as new signing Luka Milunovic scored his second goal in as many games for the club and striker Filip Kasalica also netted on his debut.
Midfielder Milunovic headed in a perfect cross by Cadu in the 14th minute before the livewire Brazilian forward converted a penalty either side of Kasalica's clinical finish after he rounded the goalkeeper.
Kasalica, who came on as a second-half substitute, found the back of the net with his first touch of the ball for Red Star to boost their confidence ahead of next week' cup semi-final against holders Partizan.
