When centre Waisea Vuidravuwalu accelerated between Elliot Daly and Andrea Masi for a soft 74th-minute try, the Aviva Premiership club looked beaten in the first of the two legs that will decide who takes the 20th and final among Europe's elite next season.
Pouring forward in a last-gasp attack, however, they forced Stade to stray offside and Goode snatched a 30-29 victory from the kicking tee as referee Johnny Lacey sounded the final whistle.
Tries from Tom Varndell, Ashley Johnson and Joe Simpson and a 15-point haul from Goode should have seen Wasps emerge emphatic winners, but they were punished for some costly lapses in concentration.
Djibril Camara and Jerome Fillol scored tries that should have been prevented shortly after the London club had crossed themselves and even Zak Taulafo's opening touch down was conceded too easily.
Stade, missing their Italy captain Sergio Parisse due to injury, will be favourites to qualify given they enter Saturday's second leg in Paris with only one-point deficit, but Wasps will take heart from their refusal to wave the white flag.
The club's medical team were busy early on as first prop Phil Swainston took an elbow in the back from Taulafo and then full-back Andrea Masi was taken out in mid-air by David Lyons.
Both received lengthy treatment and Wasps, who profited from two simple missed penalties by Julien Dupuy, must have felt aggrieved when their former loosehead Taulafo then touched down from close range with an outstretched arm.
Six points from Goode kept Wasps in touch and they went close to crossing in the right corner but Will Helu just failed to collect Elliot Daly's grubber.
Daly was proving dangerous in a strong spell for the Premiership club and their increasing dominance was rewarded with a first try in the 27th minute.
A line-out was won and the ball was shipped down the threequarters with Helu finding Varndell with a long pass and the former England wing easily rounded full-back Hugo Bonneval to score.
Stade's reply was swift and came courtesy of a passive home defence than enabled wing Camara to cross all too easily in the left corner.
The first half finished with Wasps launching a sustained attack on the right corner and the second started with what appeared to be a try for James Haskell, but the TMO decided the ball had been knocked on at the line-out.
Stade blundered again at the set-piece seconds later and this time Wasps were successful, Goode hacking the ball on for Johnson to grab and touch down.
Haskell was sin-binned for failing to retreat 10 yards at a free-kick and when Dupuy landed the ensuing penalty, Wasps' lead was down to five points.
The London club's self-destructive streak was exposed in the space of three minutes when Simpson scored a length of the pitch try after capitalising on good work from Carlo Festuccia and Nathan Hughes, only to then gift Stade five points moments later.
Simpson's clearance was charged down by Jerome Fillol and the substitute scrum-half pounced on the loose ball for a key try.
It the prelude to a thrilling climax that saw Vuidravuwalu race between a large gap in midfield before Goode intervened with the decisive act.
- Sports & Recreation
- Andrea Masi
- Elliot Daly