The two sides played out a closely-fought affair which was only decided in the 86th minute by Tyrone Barnett's seventh goal of the season.
It proved enough to hand second-placed Posh a narrow ninth victory of the season and a fifth on the spin while the Valiants remain in mid-table.
But Vale midfielder Lines, who scooped the man of the match award playing at the head of a new-look diamond formation, feels there was not much between the two sides.
"We've only got ourselves to blame really, switching off for that one moment against a team with two lads up front who can score goals," he told the club's official website.
"I was on the pitch thinking it was looking like it was going to finish 0-0 then all of a sudden, bang, that happens and it was a bit of a shock, but we proved again that we can play against one of the better teams in the league.
"You have got to take the positives out of it, there's no point in being negative, we have had a good start to the season but we just switched off for a couple of seconds and they stuck the ball in the net.
"I spoke to my mate, Mark Little, who plays for them as we walked off the pitch and he said to me 'how lucky were we?' and it's not the first time they have done that, but that's why they are at the top end of the league.
"It's not all about playing well - if they have got people who can score goals like that then they will get them wins and that is important."