Having reached the quarter-finals in each of the last two seasons, Laurent Blanc's side headed into this year's competition eager to prove themselves a powerhouse of European football.
Wednesday's Group F game away at one of the continent's more traditional leading lights always promised to be a difficult challenge for the Ligue 1 champions, and that proved to be the case as Schone rescued a point for their Dutch counterparts.
Edinson Cavani gave PSG the lead in the first half of a fixture that saw his strike partner Zlatan Ibrahimovic return to the club with whom he first made his name.
The visitors had an array of chances to make the game safe but paid the penalty for failing to net for a second time as Denmark midfielder Schone found the top corner to secure a point and delight the home fans at the Amsterdam Arena.
PSG dictated possession in the early stages and were rewarded in the 14th minute when Cavani opened the scoring with the first real chance of the game.
The Uruguay international slotted into an empty net on the rebound from Ibrahimovic's blocked effort after Ajax full-back Nicolai Boilesen had conceded possession to Lucas Moura by stumbling when attempting to clear a searching ball down the right.
Blanc's men continued to apply pressure on the Ajax goal, but were unable to craft any further openings as the home defence performed with admirable solidity.
All of Ajax's good work at the back was almost undone straight after the interval when Niklas Moisander squandered the ball immediately from the second-half kick-off, only for Lucas to fire wide with only goalkeeper Jasper Cillessen to beat.
Frank de Boer's hosts came close to capitalising on Lucas' profligacy in the 49th minute, substitute Niki Zimling shooting over the crossbar from long range before Ibrahimovic reminded Ajax of the visitors' considerable threat by narrowly missing the target at the end of a free-flowing move.
Lucas continued to be at the forefront of PSG's forays forward and would have found the goal his performance perhaps merited if not for Cillessen, who produced a diving save to tip his shot round the post after he had been played through by a superb ball from Ibrahimovic.
Cillessen was then called into action again to deny former Ajax man Gregory van der Wiel at his near post as PSG continued to spurn chances, with Marquinhos the next to miss as he headed off target.
And PSG's lack of composure in front of goal was finally punished in the 74th minute, Schone curling beyond Salvatore Sirigu, who could not keep the ball out despite getting a hand to the shot.
Schone then almost completed an unlikely turnaround three minutes later with a low drive from a set-piece that forced Sirigu to turn the ball onto the woodwork.
Blanc's men pushed again to try and seal all three points, but Ajax stood firm to hold on for a draw that should give them hope of competing in a pool that also contains Barcelona and APOEL.