The hosts came close to breaking the deadlock in the 23rd minute when Kallum Higginbotham let a throw-in run across his body before shooting from 20 yards on the half-volley, but Ben Williams was equal to it.
Doolan had an even better chance eight minutes after the restart when he found himself through on goal after a defensive error, but he hit his shot straight at the keeper when he should have scored.
Higginbotham was the next to squander a good chance for the hosts, snatching at his shot from close range to send it well wide.
Hibernian punished Thistle for their lack of ruthlessness in front of goal, taking the lead just a minute after Higginbotham's miss through Liam Craig's curling finish from an Abdellah Zoubir cut-back.
Williams was called upon twice in quick succession to maintain his side's lead with just two minutes remaining, first denying Christie Elliot from close range before tipping Ross Forbes's strike over the bar.
The hosts could not find a late equaliser in the closing stages as Hibs held out for an unlikely win.