Goode kicked four penalties, three conversions and a drop goal as Tigers grabbed a 36-23 win at Welford Road, with tries coming from Seru Rabeni, Alesana Tuilagi and Tom Varndell.
A personal haul of 21 points took Goode to 1,425 in the Premiership and ahead of previous leader Jonny Wilkinson (1,413).
"It is a nice thing to do, but it is probably not for keeps," said the 27-year-old Goode. "Jonny is still playing and there are other guys not far behind who are still playing. I just have to concentrate and keep going.
"The key this week was to get the win and play better rugby than we have in the last few weeks. I would not want to get the record and lose the game.
"We went out there with a lot of intensity and a lot of intent to play an attacking game. The scrum was immense, but it was a team performance from one to 15."
The three tries all came in the second half, though there were opportunities to get off the mark in the opening period.
"You have to be happy when you create the chances but then you are not happy if you do not finish them off," added Goode. "We have to keep working on it. Also credit to Saracens' defence for that, they worked very hard."
