Xaver Schlager gave Leipzig the lead but Leverkusen struck back with Terrier and Schick both scoring inside four minutes just before halftime.
Teenage winger Culbreath scored on his top-flight debut deep into stoppage time to round out the victory.
Leipzig fall back to fourth after a second-straight defeat, three points behind second-placed Borussia Dortmund.
"It was a really top match. It could've finished 4-4," Leverkusen captain Robert Andrich said.
"You can see how we're developing as a team."On Sunday, champions Bayern Munich need a win at Heidenheim to restore their nine-point lead atop the table over Dortmund.
The sides arrived at the Red Bull Arena undermanned, with injuries, suspensions and Africa Cup duties thinning out both squads.
Schlager gave Leipzig the lead after 35 minutes, evading several Leipzig defenders on the edge of the box before guiding a perfect shot just inside the left-hand post.
Leverkusen struck back just five minutes later. Terrier, scorer of an incredible scorpion-kick back-heel last week, levelled things up with a superb glancing header.
Four minutes later, Schick latched onto a Nathan Tella pass and shed a sliding Kosta Nedeljkovic before blasting into the roof of the net.
With Leipzig chasing an equaliser, Leverkusen went coast to coast, with 18-year-old Culbreath scoring in the seventh minute of stoppage time.

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