The Whites scored in the last minute of the game at Elland Road on Sunday to grab a 2-2 draw against Brentford after looking set to lose to the Bees.
They now head into a run of games which some fans believe they will do well to take anything from, with matches against Chelsea, Manchester City, Arsenal and Liverpool up next.
Kilgallon insists that Leeds cannot go into the games with low expectations or treating them as total free hits as if they suffer heavy defeats, starting at Chelsea at the weekend, then that will put pressure on.
“You don’t want to go there and get beat 5-0, and then it does start going, ‘well, look at Leeds now, they are struggling'”, Kilgallon said post match on BBC Radio Leeds.
“But they can go out and enjoy it, and Leeds won’t go in there like most teams and put five at the back, they will go there and play.
“And sometimes they can find it tough because Chelsea expect eleven behind the ball and to have to break them down.
“But Leeds play like they do here and Leeds do alright against the big teams”, he added.
Leeds head to Stamford Bridge on Saturday to lock horns with Thomas Tuchel’s Chelsea; they lost 3-1 at the ground last season.