The tussle between Murray and opposite number Aaron Smith lit up last week's second Test, with both scoring tries before the All Blacks won the game with a late Dan Carter drop goal.
That 22-19 defeat was a marked improvement on an opening 42-10 hammering and Kiss believes the service provided to Murray was a key factor.
"Conor, in the previous game, was on the back end of a quality of ball that was just not good enough," said Kiss. "Our breakdown work wasn't poor but it wasn't consistently effective.
"Our ball carrying and the service we gave, not just to Conor, was not as good as we would have liked. We got a bit better at the weekend."
He added: "Conor is a fantastic sponge who always wants to improve his game. What we saw at the weekend was just another layer.
"Was it a defining game for him? No. Every game is a defining moment but I think he found another part to his game and that will serve him going forward.
"I think the big challenge is to make this next game, this next opportunity, something that builds on that."
The final Test in Hamilton on Saturday is shaping up to, once again, be a battle of attrition and Kiss feels the squad has benefited from the 'inspiring' surroundings of Queenstown, where Ireland are based for five days.
They will be without Gordon D'Arcy, who shored up the midfield area in the second Test but has returned home with a calf injury, but Kiss played down the Leinster centre's absence.
Kiss said: "I wouldn't say it was anything to do with personnel and that is not taking anything away from Gordon. I think he has been brilliant for us.
"The truth is that the All Blacks are a fantastic team and the way that they open up space for their backs and their loosies (back row) is a hallmark of their game. They just couldn't get that purchase in the game the way they would have liked.
"The likes of Aaron Smith, Conrad (Smith) and Sonny Bill Williams, they couldn't quite get the game they would have liked but they just kept coming, that's for sure."
A stirring Test got the Christchurch crowd fired up, which Kiss felt was testament to his team's efforts.
He said: "We are in the coaches' box and you can't hear it all the time but when you opened the door and heard the noise, the fans were chanting 'All Blacks, All Blacks'. I haven't heard that for a long time.
"It was an intimate stadium and they certainly felt it was a ridgey didge, or a fair dinkum, Test match and they were totally engaged in it.
"They played well but they couldn't break us. It was fantastic, what the boys did, but unfortunately you want to win and that is the taste that is left in your mouth."
However, he called for a rule change concerning missed drop goals following Carter's late winner.
Kiss, who has responsibility for the Irish backs division, was unhappy that the New Zealand fly-half was gifted a second chance to land the winning points.
Referee Nigel Owens awarded a five-metre scrum to the All Blacks after Carter's first attempt at a game-clincher was deflected wide then touched down, behind the posts, by Eoin Reddan.
"If you go for a drop goal and miss then why should you get the ball back from a missed kick?" asked Kiss.
"That shouldn't have been their advantage. They went through it and missed it as we were good enough to stop it.
"It is like a tackle. If we tackled him there, trying to attempt a field goal, but we tackled him before it, we would get the ball back. But that doesn't happen. That's probably a law (the International Rugby Board) could look at."
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