Gary Warren netted a brace, including a 77th minute winner, after the hosts had twice fought back from going a goal behind.
Warren opened the scoring on 15, while Billy McKay fired Inverness 2-1 ahead after an hour.
Jason Holt (55) and Andy Webster (74) were on target for Hearts, and Butcher said: "It was a real battle and a real scrap.
"Heads could have gone down when we took the lead twice and were pegged back twice.
"But there was a lot of character shown by the boys. We have gone back to second in the table, which is lovely."