O'Leary's players compiled the sequence of victories on their way to finishing third in the top flight and another success for United in front of their home crowd would enhance the club's chances of claiming a top-six place in League One at the end of the season.
Grayson admitted that the atmosphere of Elland Road was a factor which seemed to unnerve his players when he first took charge at the end of December, but United's home ground has become a major asset in the months since the 39-year-old's appointment.
Only Leicester City and Tranmere Rovers have accumulated more points at home than Leeds this season and a struggling Swindon team have the unenviable challenge of blunting United's form next.
Leeds reached the mark of six successive victories with a 4-0 rout of Yeovil Town on Tuesday night and Grayson said: "We want to equal a record and then break a record - to keep the positive results going.
"When I first came to the club you could sense that the players were very nervous about playing at home, but we've transformed that. We look a very different team and confident in our ability."
Grayson looks set to name an unchanged line-up against Swindon with several first-team players still injured, although Andrew Hughes is available for the first time since limping out of United's 2-0 win over Cheltenham Town with a hamstring strain.
Frazer Richardson and Andy Robinson are still absent with thigh and hamstring complaints respectively.