Billy McKay opened the scoring for Inverness after only five minutes but United ensured a share of the points when Keith Watson headed the equaliser four minutes later.
Caley Thistle manager John Hughes made two changes from the side that had lost 2-1 in the Highland derby to Ross County.
David Raven and Liam Polworth came in for the the injured Marley Watkins, who had not recovered sufficiently from a dead leg, and Ross Draper, who began a two-match suspension.
Visitors United showed three alterations following their 2-2 draw against Hibernian last time out.
Watson, Brian Graham and Chris Erskine all started with Ryan Gauld dropping to the bench and neither Nadir Ciftci or Mark Wilson involved.
There was a place amongst the substitutes for new signing Farid El Alagui, on loan from Brentford, but no sign of David Goodwillie, who has been linked with a move to Aberdeen having failed to really make an impression in his second stint at Tannadice on loan from Blackburn.
Inverness were looking to end a run of three home defeats and they made the perfect start with the opening goal inside five minutes.
Danny Williams' corner was headed towards goal by Gary Warren and McKay just got to the ball ahead of United goalkeeper Radoslaw Cierzniak to turn home from close range.
The Arabs, however, responded positively and after Stuart Armstrong's powerful free-kick had been turned behind by Dean Brill, they levelled proceedings in the ninth minute.
Gary Mackay-Steven's corner was met perfectly by the forehead of Watson, who powered a header from eight yards into the corner beyond the diving Brill.
It was from another set piece by Mackay-Steven that caused panic in the home defence once more as Brill was forced to push away his corner.
The ball fell to John Rankin at the edge of the box, who saw his eventual shot deflect into the path of Graham only for the United striker to hit the post with the goal at his mercy before Josh Meekings scrambled away the loose ball.
However, a mistake by John Souttar almost gifted Caley Thistle the lead once more in this end-to-end encounter, but from the angle McKay could not find a way past Cierzniak.
The visiting custodian then foiled McKay again, this time getting fingertips to his looping header from six yards after great work on the wing by Aaron Doran as the interval approached.
Caley Thistle continued to keep up the pressure at the start of the second half and McKay glanced a header towards goal from Doran's cross, yet his effort lacked power and Cierzniak saved easily.
United were forced into a change as Graham had to be carried off on a stretcher in the 52nd minute with an injured ankle to be replaced by El Alagui.
This failed to disrupt the momentum built up by the hosts and Warren might feel he should have done better than head Williams' cross wide when rising highest at the back post.
At the other end, Brill had to look lively to spare his blushes after he let Shinnie's back pass get away from him, but he managed to hook the ball clear just before it crossed the line.
There was hardly time to draw breath as Caley Thistle were then awarded a spot-kick after Shinnie's charge into the box was halted unfairly by Rankin, according to referee John Beaton.
Nick Ross took responsibility for the penalty after scoring the last two, but his effort was saved by Cierzniak diving to his right.
It was a miss that so nearly proved costly for Inverness as Erskine rattled the upright after he had got on the end of Mackay-Steven's cross.
United then felt they had a shout for a penalty when Mackay-Steven went down inside the area under a Warren challenge as he latched on to a long through ball, but the match official was not interested.
There was one final chance in injury-time as Doran collected Carl Tremarco's hopeful long pass before turning sharply and firing in a shot from close range that ruffled only the side netting.