The Polish player came through 6-1 6-3 to set up a last-32 clash with American Madison Keys.
Radwanska dominated the first set, holding her serve well and breaking Johansson twice before wrapping it up 6-1.
Radwanska began the second set strongly, breaking her opponent's serve twice to establish a 3-0 lead. Johansson fought back with a break of her own, before Radwanska saved two break points at 3-2 to regain her two-game advantage.
The number four seed broke once more to take the set 6-3 and seal her progression into the next round.