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The Yorkshire side had the satisfaction of scoring two tries and picking up their first point of the Premiership season but Newcastle just hung on to their 100% home record thanks to May.
The Falcons conceded an early try when Steve Jones kicked aimlessly and Leeds ran the ball back with some pace and invention for Joe Bedford to step through after only nine minutes.
Three minutes later, John Rudd's determined run led to a penalty but Matt Burke's kick at goal was just wide before Geoff Parling blasted through the middle.
The Falcons should have made more of the chance, which was spoiled by a nudge forward as the pack piled in.
Jones made up for that kick with an instinctive offload to May and the winger turned the Leeds defence inside out with a 40m run to level the scores.
Leeds were under the cosh at four successive scrums and there was an argument for a penalty try when their front row stood up as they went back right on their own line but Newcastle settled for a Jones drop goal to take the lead.
And when the Falcons spurned a kick at goal to go for the corner, they were rewarded with a drive-over try from Andy Long and Burke converted from the touchline.
Leigh Hinton's penalty just before half-time gave Leeds hope at 15-8, especially when the Falcons produced some turgid rugby immediately after the break.
They paid for it when Joe Shaw and Burke put in two appalling loose kicks which led to a penalty.
Hinton made it 15-11 only for Burke to land a left-footed drop goal before the Aussie full-back saved a certain try with a touchline tackle on Tom Biggs.
Back came Leeds with Hinton's third penalty to make it 18-14, and when Eric Lund was driven over with 11 minutes left, Leeds were 19-18 up and had victory in their grasp.
But then May dropped a 40m goal to restore Newcastle's lead.
Jones missed a very kickable penalty with five minutes left and Leeds sensed an opportunity but Newcastle ended the game back in the Leeds 22 and were happy to run the clock down.
Newcastle (15) 21
Tries: May, Long
Drop: Jones, Burke, May
Leeds (8) 19
Tries: Bedford, Lund
Pens: Hinton (3)
Newcastle: Burke; May, Mayerhofler, Shaw, Rudd; Jones, Dickson; McDonnell, Long, Wilson, Oakes, Sorenson, Parling, Dowson (capt), Winter.
Replacements:Thompson, Golding, Perry, Wilson, Grindal, Dehaty, Phillips.
Leeds: Hinton; Holtby, Rock, Blackett, Biggs; Vasey, Bedford; McGee, Ma'asi, Noon, E Lund, Hooper (capt), Oakley, Clark, Lock.
Replacements: Cusack, Parkes, Bouza, Myall, Aspinall, Satala, Goodridge.
