Six of the game's eight tries came before the break as Falcons' Sonatane Takulua, Marcus Watson and Rob Vickers crossed to earn a 22-21 half-time lead.
Danny Cipriani, Christian Wade and Thomas Young replied for Wasps.
Jimmy Gopperth scored Wasps' only second-half try before Falcons' Mark Wilson earned two losing bonus points.
Wilson's quick tap penalty brought Newcastle's fourth try with just 40 seconds left, but Joel Hodgson's conversion hit the post and there was not enough time to take further advantage of Wasps being down to 14 men for the second time in the match.
The visitors spent the majority of the final 23 minutes a man short after Guy Thompson's yellow card for offside followed an earlier one for Josh Bassett's deliberate knock-on.
Wasps, who last won the title in 2008, are now five points clear of reigning champions Saracens, who visit Leicester on New Year's Day.
A stunning contest in front of Newcastle's biggest home gate of the season was lit up almost from the first whistle when Alex Tait found Takulua to score before Watson charged down the left wing to add another, Hodgson converting the second of them for a 12-0 lead.
But Wasps replied with two tries of their own inside two minutes as Ciprani collected his own chip over the top to score before winger Wade got on the end of Dan Robson's chip and chase for his 60th Premiership try.
Vickers then drove over from close range to restore the hosts' lead, followed by the first of Hodgson's two penalties, only for Young's try on his 50th Wasps appearance, converted by Cipriani, to bring the deficit back to just a point at the break.
Cipriani went off at half-time with a sprained ankle, suffered in the act of scoring his try. But, within six minutes of the restart, his half-time replacement Gopperth scored against his old club, converting his own try.
And, although Hodgson's second penalty brought it back to 24-28, further kicks from Gopperth and Elliot Daly stretched Wasps far enough to clear to ensure that Newcastle's late try was only a double-bonus-point-earning consolation.
Newcastle director of rugby Dean Richards:
"We're playing well. We're enjoying our rugby. We'd be enjoying it more if we'd won."
Wasps director of rugby Dai Young:
"Coming here is never easy. We expected a real tough game and we got one. It was always going to be a test for us. But what we showed in abundance was character and commitment.
"It was far from an ideal start for us. We've got to give Newcastle credit. They were really clinical. But sometimes it's better if it's in the first couple of minutes rather than 20 minutes in.
"I was really pleased at how we dug ourselves back into it. I was pleased that the boys didn't panic. Sometimes we lacked a little bit of brainpower.When we went six points up we were trying offloads and giving away silly penalties when there was no need to.
"We gave Newcastle a lot of momentum to come back at us. With eight minutes to go that game was still up in the air. Bu we go away with five points. We've got to be happy with that."
Newcastle: Tait; Goneva, Agulla, Socino, Watson; Hodgson, Takulua; Vickers, Lawson, Welsh, Green, Witty, Hogg, Wilson (capt), Fonua.
Replacements: Sowrey, Harris, Wilson, Holmes, Welch, Egerton, Willis, Waldouck.
Wasps: Miller; Wade, Daly, Eastmond, Bassett; Cipriani, Robson; Mullan (capt), Taylor, Moore, Symons, Gaskell, Young, Thompson, Hughes.
Replacements: Johnson, Bristow, Swainston, Rowlands, Rieder, Simpson, Gopperth, Armitage.
Sin bin: Bassett (57), Thompson (75).
Attendance: 8,730.
Referee: Tom Foley.
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Wasps went top of the Premiership as they came from 12-0 down inside four minutes to beat Newcastle with a bonus-point victory at Kingston Park.