There were close calls at either end and an unrelenting desire from two teams who both remain unbeaten after two matches of the campaign.
John Hughes' men probably shaded it on the share of chances and attacking flow but could not quite find a winning goal which would have been a fitting way to mark the 20 years to the day since the Inverness side's first Scottish League outing.
Dundee themselves could have left with full points had referee Stephen Finnie decided that Greg Tansey had handled in the penalty box late on.
The Inverness midfielder appeared to block Paul McGowan's attempted cut-back to the waiting James McPake, and the Dundee player was adamant he had handled.
"It was a definite handball," said McGowan. "James McPake was there behind their player and had a tap-in with his head so it was stopping a goalscoring opportunity. But coming up to Inverness and taking a point is a good result."
Paul Hartley, the Dundee manager, was sanguine about it. "Was it a penalty? It looked like a handball but I'm not going to moan about it. The decision was made, but our players are claiming for it. It wasn't given. We just have to get on with it.
"We had good chances in certain situations and I thought the introduction of Martin Boyle, with his pace on the right, made a difference."
Hughes was pleased with the display but is seeking more incisiveness up front. "I'm a bit frustrated we didn't win. We put in enough effort and commitment and you can see the style of play we want. We just need to be that little bit more creative in that final third, and that's where I'm looking for Russell Latapy [his No.2] to work his magic with the lads and make that wee difference."
It was the first time since December 2012 the sides had met in Inverness, a day when Billy McKay - now reportedly a target of Bundesliga 2 club FSV Frankfurt - scored a double to help the hosts to a 4-1 victory.
While Caley Thistle have retained a strong core of players from that occasion, Dundee are a team transformed beyond recognition. But one of their summer signings, the captain Kevin Thomson, lasted just seven minutes before hobbling off injured to be replaced by McAlister.
Ross Draper roused the home support soon after with a tremendous strike from fully 40 yards that dipped onto the top of the net.
The former Cowdenbeath striker Greg Stewart had a header clutched by home goalkeeper Dean Brill, while Stewart was the creator after 27 minutes when his cross invited a venomous McAlister header which was clutched superbly down low.
The hosts finished the half strongly. The ever-lively McKay wriggled into space across the front of the Dundee box after 36 minutes and fired hard and low but Kyle Letheren and the measure of it.
With a stalemate developing, Caley Thistle made an early second-half switch, bringing on Marley Watkins for James Vincent.
Perhaps the best move of the night came a minute before the hour, with the hosts unlucky not to break the deadlock. Graeme Shinnie, Watkins and Williams engaged in a rapid-fire interchange in and around the box before McPake just managed to nick the ball away from McKay's toes.
Dundee were increasingly on the defensive and Shinnie might have opened the scoring after 70 minutes when the Scotland Under-21 cap drew an excellent block from Letheren but could not control the rebound and fired over.