Heavy rain just before the scheduled start of play on the outside courts prompted organisers to shut the Centre Court roof, and Goerges went on to record a 7-5 6-7 (5/7) 6-4 win after two hours and 16 minutes.
It was a tough draw for Radwanska, who carried Poland's flag in the Opening Ceremony on Friday evening, with Goerges ranked 24th in the world.
The German enjoyed the calm conditions under the roof and she hit a string of winners in taking the first set.
The second was very tight and Radwanska pulled herself level by edging the tie-break, and she looked on course for victory when she went up an early break in the decider.
But Goerges levelled at 3-3 and, when her opponent served to stay in the match, she pounced, nailing a forehand return down the line for a clean winner.
Meanwhile, Maria Sharapova was utterly dominant under the roof against Israel's Shahar Peer, winning 6-2 6-0.
The bad weather prevented all but three other matches being completed, with world number one Victoria Azarenka among those forced out of action as her match against Romania's Irina-Camelia Begu was cancelled for the day.