The Israeli had been placed under huge pressure from Upton Park owners David Gold and David Sullivan, after a minimum three-point target from games against Fulham, Everton and Wolves was placed over the winter period.
As a result of their victory at Craven Cottage, West Ham are now off the bottom of the Premier League, and Grant was adamant he always had the faith of the club's owners despite their poor league position.
"The owners are very good with me, they haven't said anything different," said Grant after watching his side grab a 3-1 win.
"It's a free country, everyone has their opinions. The owners were very supportive despite being in difficult times.
"We want to continue with winning, our target was to win two from the three, we've won one and we are looking to continue that."
Grant believes the victory could provide a turning point in West Ham's season and, with their next game against Everton at Upton Park on December 28, feels his side may start seeing benefits from their performances.
"We don't deserve to be at the bottom but our job is to get points. I'm sure getting off the bottom will give us a lift," said Grant.
"I'm very pleased with our spirit, most important was the three points today.
"I hope it's a turning point, we want to play good football but now the target is points, we deserved a lot this season but haven't got the points to show for it.
"Today they [the players] saw the reward for the hard work they have been putting in."
Carlton Cole played an instrumental role in West Ham's victory by scoring a brace, his first in club football, and Grant conceded he may continue to start the England international in future games.
Cole's first goal came after Dickson Etuhu's poor clearance allowed the strike to capitalise and poke the ball home. The second came after another mistake, this time from Aaron Hughes, and the 27-year-old showed excellent composure to guide the ball low past Mark Schwarzer.
"Great finish by him [Cole]. He started the season as first choice, and he will want to continue that," said Grant.
"When you work hard sometimes things like this happen and you need to be awake to take a chance.
"I'm very pleased with the result because we didn't start well, we didn't create chances and they scored an early goal."