Frustration for PNE boss Alex Neil after Fulham draw
North End had led 2-0 at the interval, Jordan Hugill converting a cross from Sean Maguire, before Maguire deflected in a shot from Tom Barkhuizen for the second.
Fulham halved the deficit with a penalty from Oliver Norwood before Odoi dealt the cruel late blow.
PNE manager Neil said: "I'm disappointed and frustrated for the players, for everyone, for the fans who travelled down.
"Time was up, I had the fourth official next to me he showed me his watch and it was 95 minutes, then it started going 20 seconds and 30 seconds, the game is over and the referee allowed them to put another ball in the box.
"However, we should deal with that ball but when you stipulate a time I don't see any need to play over that time."
Neil refused to criticise goalkeeper Maxwell for the handling error which led to the equaliser.
"Listen, Maxi has been great for us, mistakes happen," said Neil.
"If it happens anywhere else on the pitch and an outfield player gives the ball away, the likelihood is nothing comes of it.
"Chris has come and collected so many crosses for us this season to relieve pressure and win us points, so I'm not going to critical of him being an aggressive goalkeeper and come to help his team.
"It is easy in hindsight now to say he shouldn't have come but if he had collected it, that would have been a great decision from him.
"We win and lose as a team, that is how it will always be."
While North End came under pressure for a lot of the second half, they had chances at 2-0 to put the game beyond their hosts.
The first-half lead was thoroughly deserved, their high-pressing game paying dividends.
Neil said: "Josh Harrop had three great chances one-on-one with the keeper to see the game off but we didn't take advantage of them.
"Sometimes when that happens it comes back to haunt you.
"We went about the first half exactly as I expected, we knew Fulham would try and force the ball through central areas and wouldn't want to resort to putting it forward.
"We hunted, pressed, grafted and ratted them all over the pitch.
"They couldn't deal with that, I've done that before here and it worked and we have done it again today.
"The problem with that was that we had a few players in the team who haven't played a lot of football recently.
"Eventually they were going to run out of steam, Ben Pearson and Sean Maguire coming off were a kick in the teeth really because the lads who came on don't play with the same energy."
"That stopped us hunting and pressing as much as we had done."
Maguire came off off before the half-hour mark with a hamstring injury, Pearson substituted in the second half having only resumed training this week after six weeks out injured.