It was also a first club goal for the former Celtic, Dundee United and West Brom midfielder, who was the outstanding Dark Blues performer.
The goal paved the way for a dominant second-half performance from Neil McCann’s men with Allan, a constant threat, taking the decisive goal superbly.
In 11 previous attempts, Dundee had never won in Inverness, with a draw and a defeat coming most recently in last season’s Premiership campaign.
Early on, the hosts looked intent to build on that fine past record. With less than two minutes gone, George Oakley broke clear behind the away defence but cut his shot wide.
It was tit-for-tat, and fast and furious, in the treacherous conditions. Sofien Moussa showed grit and desire in a couple of flashes and the Tunisian drew a strong parry from keeper Mark Ridgers after weaving into space in the home box.
After 24 minutes a clever slipped pass from Liam Polworth put Oakley in on goal before a timely saving challenge from Jack Hendry.
Six minutes later, Allan’s invention with a weighted pass had Moussa coursing into the left side of the box only to take completely the wrong choice. Rather than shoot, Moussa attempted to square to AJ Leitch-Smith and Coll Donaldson was able to knock the ball clear.
Towards the break, it was the home side who were turning the screw and, right on half-time, Dundee defender Hendry appeared to trip Aaron Doran just inside the penalty area.
To home fury, the Irishman was booked for diving.
Caley Thistle were caught cold 90 seconds into the second half.
A sweeping Dundee break saw Allan find and take a return pass from Leitch-Smith right on the edge of the box. Allan’s touch and controlled low finish into the bottom right-hand corner was excellent.
Just before the hour, the Highlanders survived an almighty scramble in the six-yard box as a Hendry strike was blocked by Ridgers and Mark O’Hara struck the bar before the home defence hacked clear.
An Allan free-kick from 25 yards was tipped over the bar by Ridgers, while the home keeper was superb in turning a Leitch-Smith strike onto the post on 73 minutes.
Desperate late efforts from the hosts saw a Daniel Mackay shot saved by keeper Elliot Parish and an Oakley header cleared by Kevin Holt, before Iain Vigurs whipped a shot just over.