Ireland winger Geordan Murphy continued his strong return from a broken leg by running in a brace of tries for the Tigers with his second coming in extra-time.
Goode put the Tigers into the lead early on after Steven Sparks was sent to the sin bin for his part in a tussle with England and Leicester lock Ben Kay.
Scrum-half Harry Ellis stretched the lead with the first of the hosts' tries after 15 minutes and Goode's conversion, followed by another penalty goal, put them into a commanding 13-0 lead.
Saracens struck back through a try to Kevin Sorrell, which was converted by Fijian fly-half Nicky Little, but two more Goode penalties put Leicester back in control at 19-7 ahead.
Murphy then ran in the first of his two tries shortly before the break to stretch the lead to 24-10.
Saracens would have even less joy in the second half.
Former Springbok prop Cobus Visagie was given his marching orders soon after the break, but Leicester failed to score during the his absence, with Goode missing his only opportunity at goal.
Little, however, pushed two opportunities wide himself and the game sparked to life again inside extra-time only, with two Saracens players – Ben Johnston and Ben Russell – being sent to the sin bin, while Leicester added insult to injury with a try to Daryl Gibson in the corner and Murphy's second of the night claiming the bonus point for the Tigers.
For Leicester Tigers:Tries: Ellis, Murphy 2, GibsonCons: Goode 3Pens: Goode 4
For Saracens:Try: SorrellCon: LittlePen: Little