Goals in either half from Byron Moore and Brad Inman sealed a much-needed win for Crewe, who had lost their last three in all competitions, conceding 11 goals in the process.
"We knew it was going to be a tough game obviously," said Grayson, whose side have now lost three on the bounce, but remain in fifth place in the table.
"Crewe have been on a tricky run recently, but they're a decent team and we know what they're capable of doing."
"The first goal was always going to be important as we felt if we got it we could exploit their fragile confidence and we hit the bar and their keeper made a fantastic save in the first 15 minutes.
"They got the goal which gave them something to hang on to. We were obviously prone to being done on the counter when we pushed people forward and, unfortunately, that was the case."
Stuart Beavon then drew another save from Phillips before Moore tucked home Chuks Aneke's clever chipped through-ball against the run of play in the 27th minute to give Crewe the lead.
Kevin Davies headed a Hume cross agonisingly wide just before the hour-mark before Crewe went up the other end to score, Inman curling home from outside the area in the 64th minute.
Crewe almost had a third as Adam Dugdale hit the crossbar soon after, but boss Davis admitted they were fortunate not to find themselves behind early on such was Preston's dominance.
Grayson added: "I think their goal made it difficult for us - we had to be patient and try and move the ball about and try and get back into the game.
"But things were going past the post for us and Crewe defended very well in the 18-yard box, getting blocks in, and on another day we could have got back into it."
- Sports & Recreation
- Byron Moore