Surrey decided to bat first at Trafalgar Road, but Kate Cross took wickets in her first two overs to leave them 10-2.
Danni Wyatt-Hodge was then out for 21, bringing Paige Scholfield and Davidson-Richards together.
They put on an impressive 143 for the fourth wicket before Scholfield was out for 87 off 82 balls.
But Davidson-Richards went on to three figures off 102 balls before she was out in the penultimate over.
Lancashire’s opening pair put on 53 before Emma Lamb was caught behind off Alice Monaghan for 20.
Eve Jones went to her third half-century of the competition, to add to one century, but she was also dismissed by Monaghan for 50.
Australian Katie Mack hit 84 off 69 balls to keep the chase on course, but when she and Ailsa Lister, for a duck, fell in successive overs to make it 214-5, it was game on with Lancashire still needing 83.
But captain Ellie Threlkeld (25 not out) provided valuable support to Morris as the two batters got those runs with two overs to spare to record the Red Rose’s third win of the competition as they jumped to fourth.
