Barcelona-born Navarro looked to be on the way to causing an upset when a single break of Radwanska's serve allowed her to take the opening set 6-4.
A victory for the Spaniard looked even more likely when she broke for a 2-1 lead in the second set.
However, world number two Radwanska stole a break of serve to reduce the arrears to 3-2, before going on to win the next four games in succession to take the match into a decider.
The Pole had now gained full control of the encounter and as a result she raced into a 5-0 lead. The next game also went the way of Radwanska, who completed the turnaround in the process.
She will now face world number 30 Jelena Jankovic for a place in round four.