The win continued one of the longest runs in Scottish football – the last time Hibs emerged with three points from a league game against City was in November 2007.
But the Edinburgh side started brightly and the longer the first half went on the more dominant Hibs, who survived an early City penalty claim became. They very nearly took the lead after 35 minutes when Siobhan Hunter and Lia Tweedie set up Rachael Small – but her effort hit the bar.
Tweedie also had a good chance just before the interval, but fired her shot wide of the target. The home side also started the second half stronger but then survived a flurry of City chances before and after the hour.
The first two fell to Claire Shine, who rejoined the club for a second stint earlier this month. The first was blocked by Hibs keeper Jenna Fife and then the striker failed to capitalise on a through ball from Leanne Ross which put her through on the goalkeeper.
In the midst of this pressure the bar again came to City’s rescue at the other end, with Lee Alexander getting the lightest of touches to ensure it didn’t go in. But the visitors’ second half pressure paid off when Crichton headed home a Hayley Lauder corner at the far post.