Goode was looking in imperious form when he was taken out of the game by a crunching tackle by the Exiles' South African captain Ryan Studwick in the 23rd minute.
Strudwick received a yellow card for his effort and Goode got eight stitches in his lip, but Saracens never recovered from the loss of their playmaker and the Irish pinched the game to leave Sarries with only one win in their last five matches.
Goode opened the scoring with a try after a beautifully-timed run onto a pass from scrum-half Morgan Williams cut the Exiles in half.
He then proceeded to add the conversion, a superb 45-metre drop-goal and a penalty.
When Sarries wing Richard Haughton kicked ahead and won the race to dive over in the corner for a try, Goode stepped up and slotted that conversion too.
The Exiles replied with a penalty and an unconverted try by Strudwick, who drove over at short range to keep his team in the hunt.
With Goode now off the field, Everitt kicked his second penalty and converted a solo try by right-wing Paul Sackey who burst through a tackle by Nicky Little – Goode's replacement – to run clear.
Little, back from World Cup duty with Fiji, landed a long-range penalty to send Saracens 23-18 ahead at half-time.
Everitt again reduced the margin to two points with a penalty early in the second-half, but Little was equal to the task and responded with a similar long-range effort.
Everitt stuck another penalty past his wilting hosts before centre Geoff Appleford broke through the middle of the Sarries' midfield to sprint 40 metres for his try.The score meant that the Exiles were finally in the lead with only 15 minutes to play.
Everitt soon added his fifth penalty of the afternoon and it looked like Saracens might not even win a bonus point for staying within seven points.
But the consolation prize was awarded in injury time when No. 8 Taine Randell sent Haughton in for his second try, Little missed the conversion attempt but it would have been of little consequence – the Exiles had stolen the day.
For Saracens:Tries: Goode, Haughton 2Cons: GoodePens: Goode, Little 2 Drop-goal: Goode
For London Irish:Tries: Studwick, ApplefordCons: Everitt 2Pens: Everitt 5