Teenage striker Tony Watt looked like capping a remarkable week with another winning goal for the champions but Hasselbaink cancelled out the young Scot's 51st-minute opener towards the end.
Skipper Scott Brown had returned to the Hoops midfield after missing the Barcelona victory due to illness but could not inspire his side to shake off another Champions League hangover after losing to Kilmarnock in their last home SPL fixture.
Perth keeper Alan Mannus easily saved regulation Charlie Mulgrew and Kris Commons free-kicks in front of a strangely subdued and sparse Parkhead crowd.
In the 26th minute, with Celtic in control, midweek goalscoring hero Victor Wanyama was presented with an opening when Commons stood a cross up at the back post but the Kenyan misdirected his header across goal, allowing Frazer Wright to complete the clearance.
Joe Ledley then headed over the bar from Commons' cross while the Scotland winger shot powerfully over soon after and it was left to Watt to break the deadlock following the restart, six minutes after being introduced for Miku.
Mulgrew's long ball was allowed to run right across the visiting defence and Watt managed to poke over the goalline to embarrass a retreating and badly exposed Mannus.
However it was the visitors who grew with confidence, despite having manager Steve Lomas sent to the stands for remonstrating with referee Iain Brines after an injury to Watt, and Hoops keeper Fraser Forster smartly saved from Liam Craig's snap shot.
Saints were level moments later though after Dave Mackay's cross eventually fell to Hasselbaink, who turned and drove past England's Forster from around 10 yards.
Celtic responded to the setback and sub Beram Kayal, on for Joe Ledley, drilled a shot from the edge of the box just over Mannus's bar, but the visitors held on for a hard-fought point in Glasgow's East End.