Second-half strikes from Ryan Gauld and Ryan Dow were enough to edge the win for United, while Liam Polworth grabbed a late goal for Inverness.
United opened the scoring four minutes after the interval as Gauld scored at the second attempt following a sweeping move from the home side.
The second goal came about from a defensive mix up when Josh Meekins under-hit the ball from Dean Brill's short goal-kick and the keeper hit the ball off Ryan Dow, who had the easy job of tapping into an empty net.
Polworth gave Caley Thistle some home nine minutes from time when he took a pass from James Vincent and crashed a brilliant shot into the top corner.
Thistle almost rescued a point in the dying seconds when Billy McKay's diving header at the back post was inches away from a goal.
In a lively start on Tayside Radoslaw Cierzniak saved a shot on the turn from Inverness striker McKay before United immediately responded with a volley from Gauld which Dean Brill pushed wide for a corner.
The visitors almost made the breakthrough when Greg Tansey played a through pass to Marley Watkins and he chipped the ball over keeper Cierzniak but Gavin Gunning got back to clear off the line.
But United squandered a good opportunity when Nadir Ciftci opted to shoot from outside the penalty area with Stuart Armstrong better placed to score.
Ciftci later crashed a low drive inches wide of the post after taking a pass from Ryan Gauld as United finished the half looking stronger.