The home side's last chance looked to have gone when Everitt missed with a left-foot drop goal attempt a the first minute into injury time.
But they managed to mount one final assault to secure their fifth successive victory in league and European competition.
It enabled their former England star Mike Catt to mark his 100th Premiership appearance – the first 91 were for Bath – with a win and a try.
Mark Mapeltoft had been man of the match in his last three London Irish games but it was his fifth-minute blunder which put his side behind.
His pointless pass was gleefully intercepted by Dave Walder who steamed home for the opener.
After Mapeltoft had made slight amends with his first penalty it was Walder's turn to have an error punished.
Irish scrum-half Paul Hodgson charged down his kick and got away with a clear knock-on before Catt kicked ahead and pounced to touch down in the 30th minute.
But Mapeltoft, who had landed 16 out of 18 kicks in amassing a personal haul of 58 points in the previous three matches, missed the conversion and a penalty attempt six minutes later.
Newcastle struck twice in first-half injury-time, first Matt Burke landed a penalty, then the Falcons pack drove Dowson over the line after they seized the ball from Irish lock Nick Kennedy at a lineout.
Mapeltoft landed two further penalties to cut the deficit to one point but Irish were still trailing going into injury time of a match dominated by an untidy forward battle and marred by too many handling errors.
They had only themselves to blame for their problems, hoisting too many kicks into the ultra-safe hands of Burke.
But surging runs by replacement forwards Roland Reid and Ryan Strudwick set them up for the final surge which secured victory.
For London Irish:Tries: Catt, HodgsonCon: MapletoftPens: Mapletoft 3
For Newcastle:Tries: Walder, DowsonCon: BurkePen: Burke