The victory was enough to give the Accies a six-point cushion over the Scottish Premier League's bottom side.
In the 38th minute of a dire match played in dreadful conditions, Hamilton striker Joel Thomas was tripped by Caley stopper Grant Munro just inside the box.
Mensing stepped up confidently to blast his spot-kick high past Caley keeper Ryan Esson to give the home side an advantage which they defended resolutely.
No one would have begrudged the Highland club a point, but their seventh SPL defeat in a row left the visiting fans unhappy and their chants of 'we want Brewster out' showed their increasing displeasure with boss Craig Brewster.
Caley had travelled south buoyed by a 3-0 Homecoming Scottish Cup win over Partick Thistle last week, but it was an attack-minded Accies - inspired by loan-signing Paul McGowan playing on the right - who started with more purpose.
However, the home side's early enthusiasm, admirable as it was, failed to manifest itself in terms of chances.
It was Caley right-back Ross Tokely who had the first effort on goal in the 10th minute when, perhaps hindered by a swirling wind, he curled a shot from 25 yards just over the bar.
Tokely was on hand to clear a dangerous cross from Thomas soon afterwards before Accies midfielder James McCarthy fired high over the bar from the edge of the box.
Accies had a penalty appeal turned down when Caley defender Pavel Mihadjuks appeared to clip the heel of McCarthy as the teenager set himself to shoot after nipping into the penalty area but referee Brian Winter waved play on.
Then, after Hamilton midfielder James McArthur was handed a soft free-kick after clashing with Munro 30 yards from goal, Accies defender Brian Easton wasted the opportunity by curling the set-piece high over the bar.
In the 35th minute, amid a deteriorating game, Inverness midfielder Roy McBain sent Don Cowie racing down the right but his cut-back was snuffed out easily by Accies keeper Tomas Cerny.
It looked like the game would trudge goalless all the way to the interval but in the 38th minute Accies were awarded a penalty when Thomas was tripped just inside the box by a clumsy Munro as he burrowed his way towards goal.
Mensing stepped up to blast the spot-kick high in to the net past Caley keeper Esson to give the home side a lead they barely deserved.
Chris Swailes had replaced the injured David Elebert seconds before the half-time whistle sounded while, for Caley, Garry Wood replaced Andrew Barrowman at the interval.
As the rain cascaded down and the wind whipped up another notch, Caley set about getting back level.
However, with the wind in their faces, the visitors struggled to get out of their own half.
In the 57th minute Wood should have made Cerny work at least when Roy McBain stood his cross up at the back post, but the Caley substitute failed to connect properly and the ball went wide before the danger was cleared.
Accies looked no closer to adding to their lead as Caley pushed harder and midfielder Filipe Morais came close in the 58th minute when he drove just over from distance.
The tiring Thomas was replaced by Lucas Akins in the 71st minute, closely followed by Adam Rooney for Cowie in what was the last throw of the dice by Brewster.
In the 79th minute Cerny made a fine save from substitute Dougie Imrie's free-kick before Accies defender Martin Canning cleared the danger.
Cerny then tipped a Morais shot from distance over the bar as Caley increased the pressure, but Accies again survived the consequent corner and the final whistle signalled another depressing day for the visitors whose SPL status is now clearly under threat.