Sharapova twice held match point but was unable to close them out and play was suspended for the day with 19-year-old Glatch holding an advantage point on her own service game.
Playing only her third tournament since being sidelined for 10 months with a shoulder injury, Sharapova had to work hard to contain a promising young player on the verge of the top 100, who mixed slice and topspin well and looked to be growing in confidence.
Glatch saved one match point at 30-40 with a rasping first serve which caused Sharapova to block the ball back long, and a second at advantage point to the receiver when she unleashed a mighty ace.
It left Sharapova ruing having let slip a lead of 4-2 in the second set, when she produced an indifferent service game and allowed Glatch to break back.
However, there was enough match time on Wednesday for India's Sania Mirza to beat 11th-seeded British number one, Anne Keothavong 6-1, 7-6 (5).
Mirza struck the ball with venomous pace from both wings, and applied such pressure during the phase in which she won seven of the first eight games that she triggered an uncharacteristic moment of fury from her opponent.
Keothavong, normally one of the calmer competitors, launched a ball in anger out of the centre court stadium and gained a code-of-conduct warning for ball abuse.
"It was a great match to play and a great match to pull out, and I am just happy to be playing on grass, which is faster,'' said Mirza. She next plays either Elena Baltacha, the Ukraine-born former British number one, or Anastasia Pavlyuchenkova, the fifth-seeded Russian.
