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Mitchell was called into action again five minutes later, tipping Lee McCulloch's effort around the post following a Gers free kick.
Rangers came agonisingly close to breaking the deadlock in the 27th minute, with Andy Little firing inches wide after being lined up by McCulloch.
Things went from bad to worse for the Gers in the 55th minute when Love's back-post cross was bundled over the line by David Hopkirk to make it 2-0.
Rangers pulled a goal back three minutes later. Mitchell kept out Francisco Sandaza's initial shot, but Little headed the rebound over the line from yards out to give the hosts a glimmer of hope.
The home side piled on the pressure for the remaining half-hour, with chances to equalise falling to McCulloch, Sandaza and Little, but Annan withstood the assault and held on for a historic win.