Ritchie de Laet had scored against his old club to put the Canaries in front at the break.
After Jon Walters saw his penalty saved following Leon Barnett's red card on the hour, it looked as if Norwich would hold on - only for Stoke's pressure to eventually tell with Jones nodding in from close range to salvage a 1-1 draw.
The key incident of the match occurred just after the hour when Walters got ahead of Barnett and into the Norwich box, before tumbling.
Referee Neil Swarbrick ruled Barnett had tripped him and, as the last man, awarded a penalty and then sent the Norwich defender off.
Walters took the spot-kick himself, but John Ruddy guessed correctly to make a fine save - as the Carrow Road faithful erupted.
Stoke, though, were to have the final say as Norwich were made to pay for some deep defending during the closing stages, which saw the visitors pour forwards which told in the third minute of added time when Jones headed in Glenn Whelan's cross.
Lambert lamented the penalty decision, saying: "It was at least three yards outside the box, so how he gives the penalty, I don't know, and then he has given the red card.
"It would have been a free-kick anywhere else on the pitch, but was not a sending off - you are getting a double hit.
"We were on the back foot against a really good side, but the effort of the lads was excellent.
"When you go down to 10 men at this level it is really tough. With 11, I think we would have seen it through."
The Norwich boss added: "You are disappointed not to win, but we are not too despondent because it was a big effort, with our first home game, I thought we played really well.
"I am under no illusions how hard it is going to be, but commitment-wise we gave as good as we got."
Lambert had made half-a-dozen changes from the opening day at Wigan, where Norwich had come from behind to draw 1-1.
The Canaries boss continued: "We are new to this level, we have to adapt to certain things, and that is what I did today, changing it a bit.
"I will need every player to perform, no matter who I pick. Whatever system we play, everyone has to go out and perform.
"I don't think I am going to get away with just playing 18 lads every week, I am going to need everybody."
Stoke boss Tony Pulis was pleased with the way his team continued to fight until the end of what has been a tough week which saw them travel to Switzerland for the first leg of their Europa League qualifier.
"We were a little bit flat in the first half, but credit to Norwich they did smashing," he said.
"I thought we should have had a penalty in the first half, for De Laet's challenge on Walters, and then before the lad was sent off we created three or four great chances.
"It was just a case of banging on the door and hoping it would open, we had to wait late until it did."
Stoke drew 0-0 with title contenders Chelsea last weekend, before their European adventure began - and Pulis accepts this season is unchartered territory for the Potters.
He said: "We are on a new journey at the football club, travelling away in Europe and having to deal with all that, then playing on Sundays again is a completely different way of running a football team in terms of preparing for games, and something we have to learn as we go along."
Stoke lost winger Jermaine Pennant to a calf problem in the first half.
Pulis hopes to add to "three or four" new players to his squad this week, adding:
"We have done remarkably well, now we need just a bit more quality into the team and that will help us this year."
Source: Team Talk
Source: Team Talk