Dundee needed a win to ease their relegation concerns but had to settle for a point when Liam Fox grabbed an 88th minute equaliser for Caley.
It was a first half in which Dundee, pumping adrenalin and playing with determination, if not class, wasted several chances.
Tom McManus smacked the base of the post with a low free-kick and Steve Lovell should have done much better in the 26th minute when he was clean through only to allow Ross Tokely to make ground and clear.
Knowing his team just had to score, Dundee boss Jim Duffy reshuffled his side and formation at the break pushing on striker Fabian Caballero for midfielder Garry Brady.
It brought immediate dividends for Dundee as they took the lead four minutes into the second half with a powerful header from Brent Sancho.
McManus's driven corner found the unmarked defender and he took full advantage bulleting in his header high into the net from eight yards.
Seeing his team struggling, Craig Brewster reckoned it was time he was closer to the action. With the clock ticking just past the hour he brought himself on. But, his team were almost two down in the following minute when Lovell was put through by Caballero. However, the striker's deft shot agonisingly came off the face of the bar.
Seeing the game slip away, Caley threw on their remaining two subs and pushed forward. Only a fantastic one-handed save from Derek Soutar in the 75th minute prevented one of them, Roy McAllister, from equalising with an excellent header into the top corner.
The Dark Blues fans urged their team on. They sang and cheered while biting their finger-nails during the gaps in their repertoire. With the final whistle approaching and the home fans sensing victory, the whole world appeared to cave in for Dundee when Caley grabbed the equaliser.
Following some sustained pressure, a deep cross from the left found Stuart Golabek in the unusual position of striker. He really should have scored but his eight-yard shot was brilliantly cleared off the line by Sancho only for it to fall to substitute Fox who whacked the rebound into the corner of the net.
It almost got worse for Dundee in the dying seconds when Brewster whipped his shot just wide, much to the relief of the home fans.