Johnson supplied eight of his points over the last 30 seconds of regulation -- capped by a pair free throws with 4.9 seconds left on a controversial call -- to force the extra frame.
The Jayhawks (24-4, 12-3 Big 12) used their fifth straight win to forge a first-place tie with Kansas State atop the conference standings.
Korie Lucious fired in 23 points and Tyrus McGee made six 3-pointers en route to 22 for the Cyclones (19-9, 9-6).
Iowa State buried a school-record 17 treys.
Celebrity Families of 2014 [PHOTOS]