The Celtic keeper gifted Dundee their goal with a wild swipe which completely missed the ball and allowed Colin McMenamin to roll the ball into the empty net.
Scott Brown and Aiden McGeady hit the visitors with two goals just before half-time to put Celtic ahead.
In the last minute, Boruc nearly handed Dundee a lifeline when he fumbled a cross on the line but there was no attacking player there to take advantage.
Celtic opened strongly and Shunsuke Nakamura's shot from McGeady's cross was well saved by the former Celt Rab Douglas.
The home side were stunned when Dundee took an early lead.
David O'Brien advanced half the length of the pitch before sending in the pass which Boruc looked to have covered.
However, the keeper incredibly missed his kick leaving McMenamin with an empty goal to shoot into.
It was a shocking error by the experienced keeper, who has had a number of such lapses recently.
Celtic tried to reply quickly but Marc Crosas' long-range shot was smothered by Douglas.
From a quick break, Scott McDonald pulled his shot wide and Brown's header from a Nakamura free-kick found the side netting.
The home team equalised when Brown cut-in from the left. From his cross, McDonald's shot was blocked but Brown latched onto the rebound before picking his spot in the net.
Celtic took the lead just before half-time when McGeady cut in from the left before firing an unstoppable shot high into the corner.
Celtic should had added to their lead in the second half but Douglas produced some fine saves to deny McGeady and Paul Caddis in particular.
McDonald missed a number of chances which he normally would have taken.
There was still time in the last minute for Boruc to cap a poor afternoon with a mistake which could easily have allowed them to equalise.