Date-Krumm took the first three games of the first set, in which Williams later recovered from a 1-5 deficit to achieve a 5-5 tally. Date-Krumm eventually went on to secure the set on her eighth set point with an 8-6 tie-break.
Williams maintained the momentum she had built up in the first set to win the second 6-3 before another close set saw the 23 seed win 6-7 (6-8) 6-3 8-6.
“It was tough,” she told the BBC afterwards. "She came out and I couldn’t seem to get a game. She just played so well. I have to give her my enormous credit. Right until the end she was attacking.”
Five-time Wimbledon champion Williams will now go on to face either Maria Jose Martinez Sanchez or Monica Niculescu in the third round.