The Eagles got off to an excellent start with Molly Sharpe nearly opening the scoring in the first 30 seconds, but her shot just went wide of goal.
Within the first five minutes Bristol City were forced into substituting their 'keeper after a shot from Millie Farrow saw Fran Bentley collide with the post.
Their sub shot stopper, Erin Foley, was thrown straight into the action, saving a close range effort from the Palace attack.
The Eagles continued to press, looking for the opener, and Under-21s midfielder Skye Bacon nearly got the first after Bianca Baptiste found her in the box, but the City defence cleared the ball off the line.
The away side had the chance to take the lead late into the first-half after striker Melissa Johnson had a shot on target, but Chloe Morgan denied it well.