Until Back's late intervention, the score had stood at 20-20 following a thrilling fightback from Saracens, who scored twice in injury-time to draw level.
Leicester looked in control of proceedings for much of the match, but as has happened throughout the season, they often lost their compusure at key moments when a score was on the cards.
The Midlanders did open the scoring through a penalty kick from Tim Stimpson, but Saracens hit back immediately following some uncharacteristic sloppy line-out play from their forwards. From a defensive line-out, the ball was slapped back over their own tryline and Richard Hill was on hand to touch down for an opportunistic try.
Goode added the conversion to make it 7-3 and that lead was cut by three points following another penalty from Stimpson.
While Stimpson eventually put his side in the lead with a third penalty from long-range, he did miss two earlier kicks from much simpler positions to keep his side under immense pressure.
Saracens actually led by 10-9 at the break but in the second half the Tigers began to stretch away from their opponents, a classic Leicester forward drive resulting in a try for hooker Dorian West and then McMullen dropped a goal.
With the clock ticking down to full-time, it appeared that Leicester had the result in the bag, but two late scores from Saracens kept both sets of supporters on the edges of their seats.
In the 40th minute, Sevens specialist Richard Haughton weaved some magic to score a brilliant individual try, converted by Goode.
Then, in the third minute of injury-time, Stimpson kicked a penalty to make it 20-17 and the match looked to be over.
The Tigers were then penalised at the re-start and Goode kicked a further penalty to draw the scores level and send the match into two 10-minute periods of extra-time.
But then Back's late try and the subsequent conversion ensured that Leicester will keep up their record of having qualified for every Heineken Cup season in which English clubs have been involved.
For Saracens:Tries: Hill, HaughtonCons: Goode 2 Pens: Goode
For Leicester Tigers:Tries: West, BackCons: StimpsonPens: Stimpson 4DG: McMullen