Everitt opened his side’s account in the sixth minute with his first penalty and then seven minutes later increased the Exiles’ lead to 6-0 after another indiscretion by the home side.
But after conceding these two early scores, the Falcons put together some good attacking phases, backrower Epi Taione effecting some impressive breakthroughs in midfield.
After working their way deep into the London Irish half, a quick tap penalty enabled centre Jamie Noon to crash over for the only try of the match.
Newcastle skipper Jonny Wilkinson slotted a brilliant conversion from wide out to give his side a 7-6 lead.
Two further penalties from Wilkinson saw the Falcons go into a 13-6 lead, but Newcastle were unable to iron out the errors from their general play with too much possession gifted to the visitors.
Everitt got his side to within a point of Newcastle by half-time with two further penalties.
The second half was a desperately close affair.
London Irish took the lead in the second-half with Everitt’s fourth penalty kick then Wilkinson put his side back in front with a penalty of his own in the 59th minute.
The final quarter saw Newcastle launching some spirited attacks, but they could not get any more points on the board, while their opponents managed another nine points with the boot.
For Newcastle Falcons:Tries: NoonCons: WilkinsonPens: Wilkinson (3)
For London Irish:Pens: Everitt (6)Drop-goals: Everitt (2)