Lesley McMaster shot Rangers into an early lead. However, second-half headers from Laura Kennedy and Jenna Ross took the trophy to Edinburgh.
Rangers have now been runners-up in two of the last three seasons.
"I am delighted for the girls and it gives them something tangible for all of their hard work for the last 10 months," said Hibs boss Willie Kirk.
"Going behind so early was not ideal, but at the time I thought it might be good for us as it would spark us into life.
"However, as the game wore on and we hit the woodwork three times, I was starting to think this was not our day.
"But the girls remained very patient and got the reward they deserved at the end. I am really delighted for them.
"They have had some outstanding performances in the Scottish Cup this year and deserve their winners medals."