Leicester were reduced to 10 before half-an-hour when Jamie Vardy was punished for a deemed lunge before Bojan marked his return to the starting eleven by converting a self-won penalty.
Claudio Ranieri's side accumulated four bookings in the final five first-half minutes as frustrations got the better of them, and Joe Allen heaped further misery on them by doubling Stoke's lead before the break.
The second-half was largely uneventful - though Kasper Schmeichel made some more fine saves to prevent his side falling further behind until Ranieri opted to bring on substitutes Demarai Gray and Leonardo Ulloa.
The latter headed in from the former's cross to hand reigning league champions Leicester a lifeline with just 15 minutes to play.
And with just two minutes to play, the visitors pulled level through Daniel Amartey as he met Christian Fuchs' cross to earn Leicester only their second point of the season on the road in the most remarkable of circumstances.
Both managers make changes from mid-week
Stoke boss Mark Hughes made three changes for the visit of the champions, two of them seeing Giannelli Imbula and Mame Biram Diouf replace Charlie Adam and Xherdan Shaqiri.
With winger Marko Arnautovic serving the first of a three-match suspension for a red card, Bojan came in for his first start in just under a month and only his fifth league start all season.
Marc Muniesa missed out on the squad entirely with a knock he sustained last weekend, although captain Ryan Shawcross made his second start in just four days after returning from a calf injury against Southampton.
Meanwhile Ranieri opted to make two changes himself, welcoming goalkeeper Kasper Schmeichel back to the team for the first time in six weeks with the Dane recovered from a broken hand.
Right-back Danny Simpson also returned from suspension to replace Luis Hernandez, although midfielder Danny Drinkwater remained sidelined with a knee injury.
End-to-end start in the Potteries
In his pre-match press conference, Ranieri spoke about the need for his side to react and improve their dismal away form, the worst record of any defending Premier League champion from eight games with just one point from 24.
The Italian was defiant that their playing style is unwavering regardless of whether the Foxes play at home or away, but his side were indebted to the returning Schmeichel inside just five minutes.
Following a sublime flowing move from the hosts, Bojan found Allen inside the box and from the by-line the Welshman managed a cute cutback to Jonathan Walters at the near post.
But Schmeichel, playing for the first time since Leicester's goalless Champions League group stage draw in Copenhagen on November 2nd, was on hand to force the forward's side-footed effort wide..
Leicester quickly grew into the game, Stoke's on-loan 'keeper Lee Grant getting down low to Riyad Mahrez's deflected strike before having to push Robert Huth's dangerous looping header from an Andy King cross over the crossbar.
Both sides took turns to go close, Islam Slimani's shot across goal underwhelming after he bullied Shawcross to meet Marc Albrighton's long ball through the middle.
Almost immediately up the other end, Bojan - having opened the scoring in his previous two appearances against Leicester - looked as though he would put Stoke ahead from a good position inside the box, only to be denied by King's block.
Hosts take advantage of Vardy red card
Stoke were then handed the initiative on 28 minutes, when Leicester striker Jamie Vardy was given a straight red card for the second time in 2016, for what appeared to be a two-footed lunge on Diouf.
Hughes' side - having had to battle to a point against Southampton at the Bet365 only a few days earlier - were now tasked with having to find space between Leicester's compact lines of midfield and defence.
But they were dealt a helping hand, quite literally, by Simpson when Bojan's cross struck the defender's flailing right hand and saw referee Craig Pawson point to the penalty spot without hesitation.
The 26-year-old Spaniard stepped up himself to send Schmeichel the wrong way, before tempers began to fray in an eventful end to the first 45 minutes.
Leicester racked up four yellow cards inside the closing two minutes, with Slimani, King, Fuchs and Albrighton all going into Pawson's notepad as the frustrations of Vardy's dismissal and the penalty seeped into their play.
It got worse before half-time too, Joe Allen tapping in his fifth goal of the season after Imbula's left-footed strike came back off the post and fell straight into the midfielder's path five yards out.
That second goal left the 10 men of Leicester with a mountain to climb, as they went in at the break trailing for the seventh of nine away games this term.
Tempo, and tempers, cool in second-half
The start of the second-half was notably calmer than the end to the first, despite former Stoke centre-back Huth picking up Leicester's sixth yellow card by the 48th minute.
The away side were kept relatively quiet as a result of their lesser numbers, Mahrez curling a free-kick disappointingly over from a promising position.
Schmeichel had to make another fine stop before the hour, tipping Walters' powerful header over his crossbar as Stoke went in search of a decisive third.
But the Potters were in cruise control and never really pressed hard to extend their advantage with the game looking won for much of the second-half.
Leicester substitutes combine for instant impact
Yet the game wasn't over as they were soon to find out. Ranieri threw on Demarai Gray and Leonardo Ulloa, replacing Slimani and Albrighton, on 72 minutes as he looked to hand his charges some impetus.
And they made the perfect contribution inside just a couple of minutes, Gray making enough space to swing a cross deep into Ulloa at the back post.
There, Shawcross was unable to prevent the striker's header from crossing the line - Pawson awarding the goal after looking down at his watch for the goal-line decision system.
That goal put some wind in Leicester's sails as they sought a second goal to complete the exact same comeback they managed here last season in September 2015, when they were also 2-0 down at half-time - albeit with 11 players on the pitch.
Amartey header earns unlikely point
They piled on the pressure looking for only their second point of the campaign outside of the King Power Stadium, and though it seemed their search would be in vain - Amartey met Fuchs' cross on 88 minutes to complete an incredible comeback and earn a point with his first goal for the club.
That means Stoke have yet to beat Leicester in the Premier League and haven't claimed a home win against the Foxes since December 1980, although they move up to 11th and Leicester drop to 15th.