Ellis has been ruled out of the entire tournament after suffering a recurrence of a hamstring injury during a domestic one-day final on March 11.
CSK moved quickly to secure Johnson, who has previously featured in the IPL for Gujarat Titans and Kolkata Knight Riders.
Johnson had initially been signed directly by Quetta Gladiators for the PSL but later withdrew citing personal reasons.
The franchise subsequently replaced him with Alzarri Joseph.
The 28-year-old has represented Australia national cricket team in white-ball cricket, playing 13 matches across ODIs and T20Is. He has taken 14 wickets in eight T20Is and will now join CSK for INR 1.5 crore ahead of the new IPL season.