The Parkhead side went two clear with goals from Robbie Keane and Aiden McGeady and it looked like the Hoops had sealed the three points.
However, there was much more drama to come.
Dons striker Steve MacLean reduced the deficit from the spot after Edson Braafheid had handled. MacLean then grabbed an 88th minute equaliser.
There was more woe for Celtic as defender Darren O'Dea was sent off in the final stages for his second yellow card following a foul on Gary McDonald.
With Rangers playing Hibernian at Ibrox on Sunday, the advantage seems to have swung towards the Govan men who can restore their 10-point advantage.
Mark McGhee, contrary to speculation, did not field a weakened Aberdeen team ahead of Tuesday's Scottish Cup fifth-round replay against Raith Rovers, which extinguished immediately any conspiracy theories suggesting the former Celtic striker was helping his old club.
However, when the visitors went on the attack right from the first whistle and raced into a third-minute lead through Kamara, it already looked like a long way back for the Dons.
McGeady worked his way in to the Dons box to set up Keane and when the Irishman's low drive from 15 yards was parried by Jamie Langfield, Kamara followed up to slam the ball past the Aberdeen keeper and into the net.
But in the 10th minute, with their first shot on target, Paton levelled with a text-book volley.
Aberdeen skipper Mark Kerr lobbed the ball hopefully into the Celtic penalty but Braafheid failed to clear properly and from the edge of the penalty area, Paton volleyed low past Artur Boruc.
An entertaining and evenly-balanced game began to unfold and with both defences nervy, more goals were always on the cards.
In the 29th minute Fortune raced on to a Paul Caddis pass but with Keane racing in to support, he opted to fire his angled-shot straight at Langfield, the Dons keeper making an even better save from Keane's close-range effort moments later.
In the 36th minute the visitors restored their lead through on-form Fortune.
Hoops skipper Scott Brown drove deep into the Dons half before laying the ball off to Keane on the right-hand side and when his pass sped across the six-yard box, the former Nancy striker stroked the ball in at the far post.
However, not for the first time this season, Celtic showed themselves to be immediately vulnerable after scoring.
A minute later the match was level again when Mackie was left unmarked at the back post to take a Jim Paterson cross on his chest and volley back across Boruc and in at the far post.
The fans enjoyed a frantic start to the second half and again it was Celtic who started on the ascendancy, with McGeady pulling the strings.
In the 53rd minute the Dons had a chance to take the lead when Mackie latched on to a long punt by Langfield to head on to MacLean inside the box.
However, with the ball bouncing high towards Boruc, the Pittodrie striker never looked balance and failed to get a decent connection, allowing the Celtic keeper to gather.
Then Langfield made a good save when he threw himself down to his right to block Josh Thompson's header from Kamara's corner.
As play raced to the other end once more, Caddis had to clear a Gary McDonald header off the line following a Paton corner.
In the 64th minute Morten Rasmussen replaced Fortune and Ki Sung-Yueng came on for Landry N'Guemo but seconds later it was Keane who headed the visitors back in to the lead.
The Aberdeen defence appeared to lose their concentration as Kamara's cross came over from the right and the Republic of Ireland striker was at the back post to head in.
After Paterson had fouled Ki 30 yards out, Langfield had to stretch to tip Braafheid's perfectly-executed free-kick over the bar but the respite was short-lived.
McGeady took Ki's corner, jinked inside the box and curled the ball past him to make it 4-2.
It looked all over for Aberdeen.
However, the excitement continued moments later when MacLean reduced the deficit from the spot after Braafheid mis-timed a jump and handled the ball in the box.
Langfield made another great save, this time from Ki before Boruc made an even more important point-blank save from MacLean.
However, with two minutes remaining, MacLean pounced inside the Celtic box to slam in a dramatic equaliser.
There was still time for Braafheid to clip the Dons bar and O'Dea to be sent off after receiving his second yellow for a foul on McDonald .