Rhinos led by eight points after 74 minutes, but London Broncos scored back-to-back tries to seal an 18-16 win and leave an 11,000 crowd stunned.
It was Rhinos’ fourth successive defeat and sixth in seven games this year.
The eight-time champions remain second from bottom in Betfred Super League, ahead of Huddersfield Giants – who have a game in hand – on points difference.
Leeds had hit back from 8-0 down at the break and Furner said: “The last six minutes was tough, very tough.
“We did a good job at half-time refocusing the group and we fought really well.
“We did a very good job to get back into the lead, That’s a positive. We were in control, but those last six minutes hurt us – all off last plays.”
Of where Rhinos go from here, Furner said: “We will have a look at it, but in this game there’s moments that define you and we have got to make sure we nail those moments.”
Furner admitted to being “angry” and “hurting”, but insisted: “I am not rattled – as I always do as a coach I will have a look at what’s [happened]. It’s a long season, there’s going to be no white flag.”
London boss Danny Ward admitted the win at his former club was “up there” with the best of his coaching career.
He said: “I thought we weren’t great in the second half, Leeds were all over us and I thought the boys looked a bit shot at times.
“I was fearing the worst with 25-30 to go, but credit to the boys. To show the resilience to come back was outstanding.”