Sam Cosgrove earned the Dons an early lead and, after being denied a penalty moments later, they looked in total control until Saints rallied late on.
The point was not enough to lift Saints off the bottom of the Premiership, where they are now level with Hamilton, St Mirren and Hearts, and third-placed Aberdeen missed the opportunity to pull further away from Motherwell.
Aberdeen stalwart Andrew Considine was handed the captain's armband on the occasion of his 500th appearance for the Dons and was afforded a guard of honour by his team-mates just before kick-off.
And they wasted little time in carving out the first opportunity of a first half they dominated.
Niall McGinn's cross from the left in the 10th minute was deflected high into the air and left to each other by St Johnstone pair Jason Holt and Ali McCann but Shay Logan's shot was just too straight to beat St Johnstone keeper Zander Clark.
But Clark had no answer 10 minutes later as Aberdeen took the lead.
Keeper Joe Lewis' punt was deflected into the path of Ryan Hedges and his lay-off found Cosgrove 25 yards from goal.
The striker moved the ball quickly onto his left foot and buried a superb drive into the bottom corner of Clark's net.
Three minutes later, the Dons should have doubled their lead.
Hedges sent in an inswinging cross from the right that tried to pick out Curtis Main lurking at the back post.
Just as the striker looked ready to get his head to the cross, defender Wallace Duffy's outstretched arm appeared to make contact with the ball.
With referee Kevin Clancy waving play on, McGinn somehow failed to find the target from the rebound just six yards out.
Cosgrove's header slipped wide in the 56th minute as Aberdeen continued to look the more likely to get the next goal after the break.
Hedges then fired over from the edge of the area after Cosgrove had robbed Jason Kerr St of possession before the match burst to life in remarkable scenes.
Saints found an equaliser with 19 minutes remaining. Kennedy was first to Kerr's pass out of defence and when substitute Chris Kane returned the ball into his path the winger fired past Lewis and into the net.
Just three minutes later, the home side were reduced to 10 men when Davidson was sent-off for an over-the-ball challenge on substitute Craig Bryson.
And within another three minutes they were down to nine men as Hendry, only on the pitch for nine minutes, lunged into Lewis Ferguson to see red.
Aberdeen tested Clark with efforts from Mikey Devlin, Considine, Ferguson and McGinn as time ticked away.
And in injury time Clark came to the rescue once more as he beat away Cosgrove's drive before Logan dragged his shot from the rebound wide.