The former England captain looked destined to taste defeat in his first home match in charge of the Hibees after Kris Doolan had given the visitors a second-half lead in the capital.
It was a cathartic finish to a frustrating afternoon for the ex-Swindon striker, having seen a goal-bound second-half shot spectacularly saved by Fox and generally endured a tough shift up front on his own.
"James has been brilliant, he has been an inspiration in the last three games," said Butcher.
"He leads the line tremendously well, works his socks off and he has good ability. He is always willing to miss opportunities and get straight back into the box again to score.
"I was absolutely ecstatic for him when he scored because it was a great finish - and finally the ball had dropped to a Hibs player in the box."
Butcher had feared it would not be Hibs' day after substitute Jason Cummings swung at fresh air when it seemed easier to tap the ball into the net from six yards.
The superb Fox also made fine saves to deny Ryan McGivern, Owain Tudur Jones and Paul Cairney.
"In the end perseverance paid off. Perseverance is the word of the day for me but for a while I thought the goal was never going to come," Butcher said.
"They scored a goal out of nothing and you could see the confidence flooding out for them. It took a while for us to get back into it, but I think there was a kitchen sink in there towards the end as we threw everything forward."
While Butcher was lauding a point achieved in dramatic fashion, his Thistle counterpart Alan Archibald conceded the draw "felt like a defeat" for his side.
The Glasgow club remain without a league win since October 20, despite large portions of their performance in Leith making a mockery of that dismal recent run.
And Archibald was left cursing the profligacy of Ross Forbes, who skewed an excellent chance wide, and some lax defending for failing to escape the capital with a victory.
"It feels like a defeat, that is all it can feel like when you concede so late. It is a sore one to take," he said.
"We will take the positives out of it. I thought our players were fantastic and worked ever so hard. We were very good throughout and it is painful to have not won.
"It could have been 2-0 when Ross Forbes had a chance but he thought there was a handball and snatched at the chance. That would have won the game for us.
"Then we lose a goal that's just down to bad defending. If he [Collins] doesn't score, there is another Hibs player waiting. There are small margins and you have to defend until the final whistle."