The British number one made a strong start to the match, getting an early break on the board before letting the German level, but Watson struck again to edge the first set.
The closely-fought contest continued in the second, but a couple of errors from Goerges left the Briton serving for the match after just over an hour on court.
The world number 62 broke back straight away though, before picking up the next game to leave her serving to level the match and force a deciding set.
Goerges was unable to maintain her challenge, fading quickly in the third to give Watson another early break, and this time the German could not respond, leaving Watson to serve it out.
Watson, who is ranked 43rd in the world, will face 29th seed Camila Giorgi in the next round.