Geordan Murphy crosses for Leicester's first try
Bourgoin: (3) 13
Pens: Boyet, Coux
Leicester: (28) 28
Tries: Murphy, Varndell, Corry
Cons: Goode 2
Pens: Goode 3
Leicester blew a disjointed Bourgoin side away in the first half to keep their Heineken Cup hopes alive.
The Tigers pack adapted well to the wet conditions as Bourgoin, without a win in the tournament, left their second-half fightback too late.
Tries from Geordan Murphy, Tom Varndell and Martin Corry swept a dominant Leicester to a 28-3 half-time lead.
Bourgoin scored twice after the break, while three yellow cards undermined the Tigers' hopes of a vital bonus point.
With the heavy rain forcing a pre-match pitch inspection at the Stade Pierre Rajon, the game was not one for fans of fluent running rugby.
But Leicester's experienced pack played the conditions to a tee from the first whistle, giving Goode the chance to knock over a penalty in front of the posts after two minutes.
Goode then made a rare outside break, before passing inside - replays suggested the ball went forward - for Murphy to dive over and extend the Tigers' lead.
Bourgoin had been virtual strangers to the ball for the opening quarter, but one excellent kick to touch got them into the Leicester 22, and Benjamin Boyet slotted a tricky penalty two phases later.
The weather conditions made for a fierce forward battle
Leicester, sticking to safety-first tactics, kept the scoreboard moving through two Goode penalties in quick succession as Bourgoin's resistance began to fade following Mark Rennie's sin-binning.
The Tigers then showed excellent handling skills to build up pressure before a flat pass from Goode found Varndell in space and the winger did just enough to score.
Shortly before half-time Harry Ellis burst into an enormous gap off the back of a ruck, and passed inside to allow Corry a stroll to the line and a 25-point lead.
The stage seemed set for Leicester to grab a potentially crucial bonus point, but their hopes were undone as Bourgoin came out with much more fight.
As Leicester's intensity level began to drop, Benoit Cabello crossed on 68 minutes to cut the gap.
Corry was sin-binned for a professional foul, and the captain was soon followed by substitute Brett Deacon for an offence at the line-out.
A deliberate knock-on from Goode also led to a yellow card as Bourgoin pounded the Leicester line late on.
And it was no less than the home side deserved when winger Jean-Francois Coux scored with the last action of the match.
Leicester boss Pat Howard:
"Most people have a tough time here - it's alway close games - so I'll take it.
"In the first half, our control was very good and we cashed in on it.
"We didn't capitalise on the momentum we had after the break and it is an opportunity lost and we'll see if it counts in the end."
Bourgoin: Laloo, Nicolas, Bousses, Coetzee, Coux, Boyet, Parra, Milloud, Vigneaux, Sourgens, Del-Fava, Williams, Bonnaire, Rennie, Monzeglio.
Replacements: Cabello, Cardinali, Khinchagishvili, Jooste, Davis, Denos, Prendergast.
Leicester: Vesty, G. Murphy, Hipkiss, Gibson, Varndell, Goode, Ellis, Ayerza, Chuter, Castrogiovanni, Cullen, L. Deacon, Moody, Jennings, Corry.
Replacements: Buckland, Moreno, Hamilton, B. Deacon, Bemand, Cornwell, J. Murphy.
Referee: D Courtney (Ireland)