Newcastle (7) 21
Tries: Gopperth, Shiels, Swinson Cons: Gopperth 2 Catterick
Wasps (10) 25
Tries: Webber, Ward-Smith, Sackey Cons: Walder 2 Pens: Walder 2
Sackey scored a wonderful try but Wasps' win was in vain
Wasps, the 2008 champions, missed the Premiership play-offs for a second year despite winning at Newcastle, after Bath's victory over Leeds.
Newcastle went ahead as Jimmy Gopperth scored a converted try but Wasps took a 10-7 lead through Rob Webber's try and five points from Dave Walder's boot.
Grant Shiells scored for Newcastle but Dan Ward-Smith replied quickly.
A Paul Sackey try and Walder penalty sealed the win before Tim Swinson got a late consolation for the Falcons.
Wasps entered the penultimate round of fixtures in control of their own destiny but losing to Bath at Twickenham gave the West country side the whip hand and they duly clinched the fourth play-off spot with a big win over Leeds.
Bath's win also means the Wasps careers of Danny Cipriani, who left the field early on with a suspected fractured thumb, and Paul Sackey end in disappointment before their moves to Melbourne Rebels and Toulon respectively.
Newcastle captain Carl Hayman also made his final appearance for his club before joining Sackey at Toulon.
Defeat sees Newcastle, for whom lock James Hudson was outstanding, finish a lowly 10th but, in head coach Alan Tait's first game in charge, there were encouraging signs their fans can look forward to a better campaign next season.
Tait had only been handed the reins following the departure of Steve Bates on Tuesday, and he could hardly have wished for a better start.
Gopperth, the Premiership's top points scorer, cut against the grain and brushed off a weak Cipriani tackle to cross for a third-minute try he also converted.
Cipriani was then forced off in the 23rd minute after injuring his thumb in a tackle shortly before dotting down a Newcastle kick, with Mark van Gisbergen his replacement.
And the visitors shrugged off that blow to level the scores as a well-controlled driving maul allowing Webber to burrow over the line, Walder adding the conversion against his former club.
Walder then gave Wasps the lead as he knocked over a simple penalty before Wasps were reduced to 14 men in the 48th minute when Webber was sin-binned for pulling back the Newcastle support after Hudson had taken a Gopperth cross-kick
They were punished immediately as lock Swinson offloaded out of the tackle to send replacement prop Shiells over, Gopperth judging the blustery conditions to perfection to land the conversion.
However, the game turned back in the visitors' favour three minutes later after Gcobani Bobo's poor clearance kick was charged down and Ward-Smith crashed through a tackle to cross, although Walder failed to add the extras.
Newcastle laid siege to the Wasps line but were unable to make their pressure tell and they were punished in the 63rd minute.
Simpson used his pace to make an intelligent blindside break from the base of a scrum on halfway and freed Sackey on the right.
The England wing side-stepped the covering Alex Tait to score a wonderful try which Walder converted, before adding his second penalty to give Wasps a winning advantage.
Newcastle kept going and had the last word as Swinson barged over in the final seconds for a consolation try, converted by replacement Tom Catterick.
Wasps head coach Shaun Edwards:
"I don't think it was the Bath game that cost us - there were other fixtures throughout the season where we were decimated by injuries.
"We had to go into certain games with our fourth or fifth choice tight-head prop and, looking back, some of those games have cost us a play-off spot."
Newcastle head coach Alan Tait:
"In the first half we missed some opportunities and it cost us - it's just about the back three getting on the ends of good bits of play but we haven't quite been sharp enough in that area.
"But we certainly took it too them. Wasps are a very good defensive side and there were times we had them stretched all over the place.
"We are proud of them and we can definitely take plenty from that and there is enough in that game to take us forward and I am looking forward to pre-season and starting again next season."
Newcastle: A Tait; Williams, Vickerman, Tu'ipulotu, Bobo; Gopperth, Young; Ward, Vickers, Hayman, Hudson, Swinson, Levi, B Wilson, Afu.
Replacements: Davey for Williams (70), Catterick for Gopperth (79), Charlton for Young (45), Shiels for Ward (41), Thompson for Vickers (70), Brooks for Hayman (79), Sorenson for Afu (61).
Not Used: Welch.
Wasps: Cipriani; Sackey, Jacobs, Waldouck, Varndell; Walder, Simpson; Payne, Webber, Vickery, Ward-Smith, Skivington, Betsen, Rees, Hart.
Replacements: Van Gisbergen for Cipriani (21), Taulafo for Payne (63), Lindsay for Webber (75), Broster for Vickery (60), Leo for Rees (73), Matthews for Hart (61).
Not Used: Honeyben, Lemi.
Sin Bin: Webber (48).
Ref: Rob Debney (Leicestershire).