Harry Leonard scored a 94th minute winner as Posh secured a first Sky Bet League One victory of the season.
Peterborough United made it two wins from three in all competitions as Luke Williams' side overcame Mansfield Town 2-1 at the Weston Homes Stadium thanks to a last-gasp winner from Harry Leonard.
It was a deserved success with Posh the better side for long periods in the game, but it would be remiss to ignore two decent opportunities for the visitors in the first 45 minutes.
George Nevett showed excellent awareness to clear off the line from a poked finish from Jon Russell, while goalkeeper Alex Bass did brilliantly to keep out Rhys Oates.
Brandon Khela had the first chance of the game for Posh on 11 minutes, but his left-footed effort was routine for Roberts after good work from the lively Owura Edwards.
Bass did well to repel Russell at the near post on 23 minutes before Ethan Williams sent a rising drive over the crossbar from long-range. Smith headed wide moments later before Bass produced the excellent stop to deny Oates.
Khela tested Roberts on 39 minutes with a stinging drive before Woods curled wide from 25-yards after two successive efforts were blocked.
Moments after the re-start, Kelly wriggled into some space in the box before curling inches wide of the target before the deadlock was broken as WOODS timed his run into the box to perfection, tapping home after Leonard's shot deflected into his path.
On 58 minutes, Mansfield had a goal chalked off for offside as Oates tapped home at the back post before SMITH levelled the scores, tapping home after a mistake in the Posh back-line.
On 65 minutes, Posh felt they should have been given a spot-kick as Johnston went down in the area, but the Northern Irishman was booked for diving. Akins stabbed wide on 73 minutes before Posh appealed in vain for another spot-kick as Williams appeared to be upended in the box.
Four minutes into stoppage time, Posh got the winner as substitute Bolu Shofowoke danced past a few defenders before finding LEONARD who made no mistake from around eight yards.
Posh travel to Watford in the second round of the Carabao Cup on Tuesday night before travelling to Blackpool next weekend.
Posh: Bass, Johnston, Weir, Kelly (sub Jones 73min), O'Connor, Khela, Leonard, Williams (sub Masara 73min), Woods (sub Conn-Clarke 69min), Edwards (sub Shofowoke 67min), Nevett, Unused subs: Smith, Mills, Dornelly,
Mansfield: Roberts, Oshilaja, Blake-Tracy (sub Moriah-Welsh 77min), Sweeney, Reed, Akins, Russell (sub Hendry 68min), Thompson (sub Maris 85min), McLaughlin (sub Dodgson 68min), Oates, Smith (sub Hewitt 68min).
Half-time: Peterborough United 0-0 Mansfield Town
Full-time: Peterborough United 2-1 Mansfield Town
Referee: Martin Woods
Attendance: 8,900 (1,598 Mansfield Town fans)