"We believed that we could win the game, [maybe] not 4-1, because Southampton played very well tactically in the first half," Grant told ITV1.
"Second half we were much better."
The Israeli felt the result was just reward for the long-suffering fans of the beleaguered club.
"It's good for the club and good for the fans," he said.
"Since the draw everybody spoke only about this game, like there is no league, so it is very important for everybody.
Keeper David James predicted the troubled club would avoid closure.
James told ITV1: "It was a wonderful day for the FA Cup and Hampshire.
"The performance on the field by both teams made it a good game, so 'happy days'."
With the club having been granted extra time to fight a winding-up petition, James said he expected Portsmouth to be saved.
"My instinct says it will be," he said. "I think it's too important for too many people, for football even, that a top-flight club can go out of existence.
"It's not going to be easy and whoever it is who helps us out we'll thank them now, but it's important we stay in business and important Portsmouth keep flying the flag in the Premier League for Hampshire."