Ross also scored in added time as the lead changed back and forth in a thrilling Scottish Cup fourth round tie which will have to go to a replay.
Billy McKay had put the visitors ahead at the Global Energy Stadium on the half hour only for Rocco Quinn to equalise four minutes into the second half.
Foran put Caley back in front with 16 minutes to go back County were level again when Iain Vigurs netted with four minutes left on the clock.
Richard Brittain put the home side ahead for the first time in the second minute of added time but Foran ensured the spoils were shared.
Aaron Doran's cross from the right was headed down by Foran and McKay swept a stunning left foot volley past Brown from 14 yards.
The visitors were well worth their lead and Andrew Shinnie almost added a second a minute later when he fired into the side netting following a corner.
There was action at both ends in first half injury time. Antonio Reguero pushed out a Brittain corner and Morrow's low drive was cleared off the line by Foran.
Caley Thistle broke quickly and McKay rounded Brown only for Mihael Kovacevic to get back to scoop his shot behind. From the resulting corner Foran headed wide of the far post.
County mounted the first attack of the second half and the ball broke to Vigurs on the edge of the area but his shot was straight at Reguero.
But the home side's early pressure paid off and they equalised in the 49th minute. Morrow squared the ball to Kovacevic on the right and his cutback was blasted high into the net by Quinn.
County were looking much more up for the fight and after Brittain and Vigurs linked on the left Kettlewell got on the end of the latter's cross but Reguero held his effort comfortably.
The visitors weathered the storm and they threatened again in the 70th minute when Andrew Shinnie tested Brown with a 25-yard shot but the County keeper got a hand to it.
But the Inverness side took the lead in the 73rd minute when a McKay shot deflected off Foran and into the net.
Just when it looked as if the visitors had the tie in the bag when Vigurs fired home a magnificent equaliser from 25 yards which flew into the net.
A Brittain free-kick in injury time evaded everyone on its way into the far corner of the net.
County looked to have booked their place in the next round but Foran popped up with an equaliser in the fifth minute of injury time when he scooped the ball into the net from close range following a Graeme Shinnie corner.