The Lilywhites weren't at their best at Deepdale but won 2-1 thanks to goals from Paul Gallagher and Jayden Stockley.
Gallagher put them ahead early on, turning in the rebound after Hatters goalkeeper James Shea had parried his penalty.
James Collins equalised for the visitors from the penalty spot at the end of the first half, Paul Huntington harshly pulled up for foul on Callum McManaman.
The second half proved frustrating for North End until substitute Stockley struck in the 84th minute.
Neil said: "The biggest thing is we won't be talking about this game in a week's time, we will be talking about going third.
"You have to understand that Luton are fighting and scrapping for points all the time.
"Our biggest error of the day was letting them back into the game.
"We got the first goal and that was the breakthrough we needed against Luton, make them come out to try and equalise and hurt them.
"It didn't happen that way, I thought we were a bit slow in the build-up, a bit sloppy. Our two wide players rolled inside too much which meant the full-backs had to go really high to offer us width.
"That left spaces down the sides for them to counter."
It was a counter attack which led to Luton's 43rd minute leveller, Neil unhappy with the decision to award a penalty.
Neil said: "They countered after Patrick Bauer misplaces a pass and it ends up in our box.
"I thought the penalty was really soft, I have seen it back and it was a poor decision but we allowed him to make a decision by being sloppy further up the pitch.
"When it went 1-1 we didn't have enough runners through the middle going beyond, we played in front of them for a lot of the game.
"It is tough when teams sit in their shape, don't move and hit you on the counter.
"We tried to disrupt that. I brought on Josh Harrop as a more technical No.10 but that didn't really work because the spaces weren't really there.
"I then went with proper wingers to hold width and give us two up front. We got our goal off the back of it."