The Highlanders looked dead and buried at the interval after conceding first-half goals to Arvydas Novikovas and John Sutton and also seeing striker Richie Foran's sent off for a reckless challenge on Ryan McGowan on the stroke of half-time.
However, the visitors staged a superb-second half comeback to salvage a 2-2 draw, Andrew Shinnie reducing the deficit from the penalty spot before teenage substitute Conor Pepper fired a dramatic stoppage-time equaliser.
A delighted Butcher said: "Honestly, it feels like a win for us, it really feels like a win. We are two points ahead of where we were last season, still unbeaten. What a feeling.
"I feel that was one of the best performances by an Inverness side since I have been here.
"It's hard to pick out one player, it was just a terrific team effort. The player's just don't know when they are beaten, there is a terrace spirit and the boys don't know when they are beaten.
"We had to ride our luck, there is no doubt about that, but the boys deserve tremendous credit.
"We had roll up our sleeves and I think this result sends out a message to everyone: this is a team which will be hard to beat."
On Foran's dismissal, Butcher said: "I spoke to the referee at half-time and he told me it was a reckless challenge and a red card.
"I disagree, but that is how he saw the incident and we had to adapt. We now face being without Richie for the game next week."
Pepper, 18, ensured a debut to remember with his late, late show. He only made his first start for the Under-20 side last week, but became an instant hero with his dramatic equaliser on Saturday.
However, Butcher revealed he feared the youngster had fired his effort over the bar.
"If the goal had been another two yards further back then I think he would have put it over the bar. That would have sent me on a pub crawl through all the Fringe bars in Edinburgh," the former England captain joked.
"Conor has just joined the club, he is on absolute buttons at the club, but his heart is as big as a bucket.
"But I'm sure he thought he was playing Gaelic football, trying to hit his effort over the bar."
Hearts boss John McGlynn was more concerned by his own side's failings, as he saw a seemingly comfortable advantage disappear.
The 50-year-old also had the added setback of seeing skipper Marius Zaliukas limp from the field on the stroke of half-time.
The Hearts skipper is now a major doubt for Thursday evening's glamour Europa League tie against Liverpool at Tynecastle.
McGlynn said: "We were keen to get the third goal, we had opportunities to do that, but lo and behold, we show some naivety late on and the ball ends up going across the face of goal and ends up in the net.
"We have massive game on Thursday so there is no point crucifying people. Everyone is disappointed and knows we should have won the football match.
He added: "Marius has a dead leg, high up on his thigh. It doesn't look good, although it is very early at the moment.
"If it is severe then I wouldn't think he will have a chance of playing on Thursday, but I have not had that confirmed by the physio."
