The Yorkshire side head to the Emirates Stadium tomorrow night to face Arsenal in a repeat of last season's third round tie in which they were minutes away from causing a shock as the Barclays Premier League side needed an injury-time equaliser from Cesc Fabregas to earn a replay, which they went on to win 3-1 at Elland Road.
That impressive effort against Arsene Wenger's side last term came a year after Grayson's side pulled off a shock by beating Manchester United 1-0 at Old Trafford in the third round as an npower League One club.
They followed that by taking Tottenham to a replay in the fourth round by drawing 2-2 at White Hart Lane, though were again undone in the replay on home soil.
With Monday's game the third successive time they have travelled to top opposition in the third round of the competition, Grayson believes the stature of United, starting with their hoards of boisterous fans, make it easier for his side to compete.
"Sometimes it just fits with the football club. Everybody talks about us being a big football club and sometimes it just fits that we can handle ourselves in the big arena, not just the players but the football club in general," he said.
"The supporters go to any ground in the country and people recognise the fantastic support we create and the noise we make.
"I took my son to Manchester City versus Liverpool the other night and he said 'will they make as much noise as Leeds fans do?' and I said 'no' and they didn't.
"We are not nervous when we go into these games because the players feel comfortable in the atmosphere that they are involved in over the course of the season by being a Leeds United player anyway.
"Obviously there is sometimes a difference in the quality of players but hopefully that is just the difference at this moment in time and in the next couple of years we will be able to get back to that same sort of level."
Following their relegation from the Premier League almost a decade ago, Leeds have regularly been on the wrong end of results at smaller clubs and Grayson is adamant games like Monday are a natural stage for the club.
"It is a privilege to play for Leeds United and a privilege you have to be able to handle.
"Those supporters demand certain things from you and maybe sometimes the bigger games bring out the best in the players.
"It is the same group of players who will, a few weeks before or a few weeks after, go to a smaller club and get beaten because sometimes it is hard to pick yourselves up mentally.
"Sometimes the football club doesn't feel like it should belong in a League One or even the Championship to a certain degree.
"But we have to earn the right to get out of those leagues."
Leeds go into the game having ended a run of three successive defeats, but only just as they needed a late turnaround against 10-man Burnley last Monday to secure a victory which in many people's eyes kept Grayson in his job.
And despite getting carried away in the emotion of that late success at Elland Road, the 42-year-old insists he never lost faith in his ability to do the job.
"I wanted to stop the rot from my own point of view but more so for the club and the supporters.
"You create your own pressure - I don't get too down when we lose and I don't get carried away when we win and I have just carried on to do the job to the best of my ability and have belief in myself."