United came into the match hoping to move within four points of their third-placed opponents in the race for UEFA Europa League qualification, but suffered a significant early setback as Szromnik received a straight red card for bringing down Marley Watkins in the area.
Replacement keeper Radoslaw Cierzniak's first task was to pick the ball out of his own net as Greg Tansey converted the resulting penalty.
However, although Inverness dominated the remainder of the opening half - with Watkins a constant threat - United responded strongly after the interval and salvaged a draw through Ryan McGowan's deflected 76th-minute strike.
Tansey almost put Inverness back in front late on, but his free-kick was tipped over by Cierzniak.
Inverness were then reduced to 10 men themselves at the death as Gary Warren saw red following an off-the-ball incident involving Nadir Ciftci.
John Hughes' Inverness are now four points adrift of Aberdeen in second, with United a further seven back.
