The result ensured Bristol fell out of the race for a play-off spot and set up a shootout between Exeter and Saracens on Saturday for a place in the top four.
Third-placed Leicester travel to Bath on the final day with a home semi-final in the play-offs on offer for the winner.
Northampton secured top spot with a 36-32 win over Gloucester on Saturday.
"Going into these games, you get that little bit of flavour that if you're going to win this tournament, you've got to be able to beat these teams," said Exeter's forwards coach Ross McMillan
"For us now, our focus shifts towards Saracens next week."
Just one point separated the sides at half-time as Woodburn's early try was cancelled out by George Pearson and Leicester's Orlando Bailey kicked three penalties to Henry Slade's two.
Exeter took control early in the second period as Len Ikitau crossed before setting up Woodburn for his second score.
But Woodburn was then yellow-carded for a deliberate knock-on and Olly Cracknell forced his way over to reduce the Tigers' deficit.
Leicester went back in front 11 minutes from time when Charlie Clare crossed for their third try.
Exeter hit back and secured both victory and a bonus point when Andrea Zambonin went over in the left corner.
"It means nothing in terms of what we have to do next week, that stays the same, but it's incredibly frustrating," said Leicester head coach Geoff Parling after a first home defeat since March 2025.
"That's not what we want to do in front of our fans. So we've got six days to go at Bath. We will review this, but we'll dust ourselves off very, very quickly and we know the task at hand."