Aaron's Doran's sixth-minute opener had the hosts in front at the interval.
But Elliott levelled matters in the 54th minute to ensure Hearts made it eight games unbeaten against Caley Thistle.
Inverness defender Ross Tokely was given his marching orders for a second yellow card 12 minutes from time, but the home side comfortably held on to claim a deserved point.
Source: PA
Source: PA