Keothavong sent packing from Stanford
Anne Keothavong was bounced out of the Stanford Classic in the first round after a 2-6 7-5 6-4 defeat to Mallory Burdette of the US.
Britain's number one women's player wrapped up the first set without too much fuss and looked set to join compatriot Heather Watson in the next round.
But a valiant fightback from Burdette saw the 21-year-old American unsettle Keothavong and claim the second set.
Burdette managed to hold off Keothavong in the decider and move into the second round of the Bank of the West Classic where she will meet second seed Marion Bartoli.
Urszula had to watch her older sister contest the Wimbledon final on television on Saturday, and while it was an emotional day watching the new world number two lose to Serena Williams in three sets, it did not deter her from own task on Tuesday.
She struggled for most of the match against the veteran Daniilidou, but in the end was the more aggressive and accurate player on the big points.
The 20-year-old will face eighth seed Marina Erakovic in the second round after the New Zealander beat Jana Juricova of Slovakia 6-2 6-2.
Romania's Sorana Cirstea, seeded ninth, progressed with a 7-5 6-4 win over home hope Vania King, while the day's other winners were Zheng Saisai, Erika Sema and Nicole Gibbs.
Wimbledon champion Serena Williams kicks off the defence of her title on Wednesday against local girl Gibbs.