After a 5-0 hammering at Artmedia Bratislava in midweek, Celtic looked well placed to restore some pride when they led 3-1 at half-time thanks to a John Hartson hat-trick.
However, Terry Butcher's men battled back after the break and the visitors needed a last-gasp strike from Craig Beattie to rescue a point.
After a feisty start which saw young Celtic defender Stephen McManus lucky to stay on the park and Motherwell losing skipper Scott Leitch to injury, the Bhoys opened their scoring account under Strachan in the 14th minute.
Alan Thompson managed to nick the ball past Alan McCormack and send a cross to the far post where Hartson rose to head home.
Six minutes later it was all square when Well equalised after fine move involving Brian Kerr, McCormack and Paul Quinn finished with Kerr bending in a left-foot shot from 12 yards.
Celtic were not to be denied and were back in front after 32 minutes when Hartson headed an Aiden McGeady cross past Gordon Marshall.
Hartson completed his hat-trick a minute before the interval when he converted a penalty after Gordon Marshall was adjudged to have tripped Stilian Petrov.
That should have put the game beyond the home side. But, Motherwell roared back into the game in the second half and were level within 15 minutes.
Firstly, Jim Hamilton headed a McCormack cross beyond David Marshall and then Scott McDonald repeated the feat from a Jamie Paterson centre.
The home side took the lead with six minutes left when William Kinniburgh turned a Hamilton cross past the stranded goalkeeper.
However, the visitors stormed back and levelled in the 90th minute when Beattie latched on to a long ball and poked it past the advancing Gordon Marshall.
Home boss Butcher had mixed feelings after the match.
"The players are upset as we have lost a last-minute goal and disappointed we haven't taken all three points," said the former England captain.
"But, from 3-1 down you are delighted to take a point at the end of the game.
"The players at half-time were having a go amongst themselves as they were disappointed at losing three goals and I had to calm them down."
His Celtic counterpart Strachan was more philosophical and joked: "I think the subscriptions for Setanta will go up with games like that but as a manager you don't want to watch.
"It was a concern to lose the goals we did as they were similar and it needs to be sorted."