The Yorkshiremen have struggled badly since returning to the top flight but they ground out a famous success against a Falcons side consisting of England quartet Jonny Wilkinson, Jamie Noon, Toby Flood and Mathew Tait.
It was only Leeds' second win in 17 Premiership outings this term and survival remains a long shot with only five games remaining.
But, if they do go down, they will go down fighting and their spirit once again came to the fore in an outstanding team performance which saw them set a new record for the fastest try in Premiership history.
Little-known centre Lee Blackett, 25, crossed the line after just 8.28 seconds to beat the record previously held by Tom Voyce, who touched down for Wasps against Harlequins after 9.63 seconds in 2004.
Lancaster said: "We've picked up early tries in most games but we've not managed to hold on.
"Our second-half performance was probably the best this season and that ultimately gave us the win that we deserved.
"We need two or three more wins but it's one game at a time and that has always been the mentality.
"We knew we needed to win tonight - it was a must-win game - and I'm proud of the players to be honest.
"I'm just pleased that they can go and have a beer tonight and have a weekend off before we travel to Saracens next Sunday."
Try-scorer Blackett - who touched down straight from fly-half Alberto Di Bernardo's kick-off - said: "I said to Albie to give me the ball inside the opening minute but I didn't actually mean within the first eight seconds!
"Obviously, it was a great start and, like Stuart said, it was an impressive performance as a whole.
"Amongst the squad, we have never, ever thought we are down.
"Obviously, if we had lost tonight the gap would have got even bigger but I genuinely, 100% believe that we've got a chance of staying up this year.
"I think Stuart and everyone in the squad genuinely believes that.
"People in the media and other people outside will write us off but we want you to write us off - we're happy for you to write us off."