A goal seven minutes into each half gave the Posh a place in the fourth round at London round, Martin Samuelsen and Conor Washington sending United through to the fourth round.
On loan West Ham man Samuelsen opened the scoring after seven minutes, fizzing a shot into the bottom corner from 25 yards and the same time after the break Washington headed home to double the advantage.
North End made four changes to the side that defeated Rotherham United the previous Saturday, Chris Kirkland making his debut in place of Sam Johnstone, who was unavailable due to the terms of his loan deal from Manchester United; Josh Brownhill also into the side.
The home side started brightly, Alan Browne clearing a Connor Washington effort in the first minute, however, the first clear cut chance came when Eoin Doyle was in on goal from Joe Garner’s flick on, but he couldn’t bring the ball under complete control and Ricardo Santos recovered to clear.
However, the Posh grabbed an early lead when loaned Hammer Martin Samuelsen picked up the ball and shot from 25 yards into bottom right hand corner of the goal.
They tried to add to that lead with another long distance effort, Harry Beautyman shooting from the edge of box only to see Kirkland tip over the bar, after the No.16 collected a flick on from Washington.
Just after the half’s mid-point Doyle was again clean through from this time from a Welsh flick on, but he drove his shot across keeper Ben Alnwick but over the bar.
The chances now were now all for North End, firstly a Joe Garner header from Adam Reach’s deep cross was tipped over by Alnwick and from the resulting corner Paul Huntington headed wide of the right hand post.
A Josh Brownhill free kick with a lot of dip was just over the cross bar from 30 yards just after the half hour mark, but in the three minutes of first half injury time there were two more gilt-edged chances for the Lilywhites.
Firstly a Reach corner from the left was headed up in the air; it dropped for Alan Browne, but he scuffed his shot just wide of the right hand post. Then with seconds left a Garner shot from the edge of the box was deflected by Jack Baldwin up in the air for United skipper Alnwick to claim.
North End were forced into a half-time change, John Welsh looked like he had taken a knock early on and battled through until the interval, but was replaced at the break by Pau Gallagher.
The second goal, like the first, came seven minutes into the half, a cross from the right by Smith was headed expertly into the top left hand corner by top scorer Washington from 15 yards out.
There was almost an immediate reaction as Doyle’s effort from Alan Browne’s left wing cross was smothered at the second attempt by Alnwick.
Joe Garner was then found at the back post by Doyle as North End continued to press to get a goal to bring them back into the tie, but his volley from the left wing ball was into the side netting from close range.
Callum Robinson was introduced for Brownhill with 34 minutes still left, his third debut , after joining on a season-long loan from Aston Villa earlier in the week.
North End were now throwing numbers forward and putting a lot of pressure on the home side .Two corners came and went, the latter headed over by Tom Clarke as the game entered its final third.
The pressure was incessant and Reach fired over on 64 minutes after some excellent possession play and build up, before Robinson fired over at the back post after seconds later.
The next chance was a Garner header from Gallagher’s corner, but it smashed against the post behind the goal, before the home side showed they were dangerous on the counter, Kirkland saving well from Washington in a rare foray forward by Posh.
After a rare spell of pressure from the home side just before the last 15 minutes of the game, the second half returned to the same pressure being exerted by the side in yellow, Robinson and Reach both looked to stretch the defence.
Tom Clarke’s header from Adam Reach’s corner was brilliantly headed off the line by Smith when it looked like the deficit had been reduced as the game entered its final ten minutes.
Callum Robinson again looked like he had brought North End one goal away from parity with a superb turn and shot from the edge of the box, but this time Baldwin was the man to head off the line and deny the No.37 a goal.
It was chance after chance, as Hugill headed goalwards, only for Alnwick to claim, after Robinson had found the substitute with a cross from the left.
Five minutes were added on, but North End couldn’t get the goal that the chances they created deserved and their FA Cup campaign for 2016 came to an end.
Peterborough United line-up: Alnwick (c), Smith, Baldwn, Oztumer (Maddison, 72), Santos, Washington, Beautyman, Angol (Coulibaly, 73), Forrester, Samuelsen (Fox, 90), Wilson. Subs not used: Henry, Anderson, Lopes, Addison.
PNE line-up: Kirkland, Vermijl, Reach, Clarke (c), Huntington, Cunningham, Brownhill (Robinson, 56), Welsh (Gallagher, 46), Browne, Garner, Doyle (Hugill, 75). Subs not used: Hudson, Wright, Kilkenny, Woods.
Attendance: 7,665 (TBC away fans)
Referee: Mr T Harrington