Lee Holmes put North End ahead seven minutes before the break and Nicky Wroe doubled the advantage after 78 minutes after pouncing on goalkeeper Eirik Holmen Johansen's error.
Jack King sealed the win for the Lilywhites in stoppage time to leave them 10 points in front of fourth-bottom Scunthorpe, who have just four matches remaining.
"It's a win and three goals for the team that have been coming over the last few weeks," said Grayson, whose side have five games left to play.
Wroe's goal put Preston firmly in command and it all came from goalkeeper Johansen's poor clearance with the midfielder scoring from 40 yards out.
For Grayson it was the decisive goal and after reprimanding the 27-year-old for trying something similar earlier in the season, he was happy to be proved wrong.
"The second goal was a big one and kills the game off, then King got a goal and how we didn't get a couple more is astonishing," said Grayson. "I berated Nicky a few weeks ago for trying to score from the halfway line. He glanced over to me when he did score so full credit to his confidence and having a go.
"I've wanted the strikers to score. Jack's missed a couple of chances over the last couple of weeks so that'll do him the power of good.
"Any manager will say we're not mathematically safe but we've given ourselves a good opportunity, if we win the next one that'll secure our position regardless of what others do."
Grayson added: "Keith Keane's been struggling with a hernia and will have an operation sooner rather than later but you see, with or without the armband, he leads by example."