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Steven Gerrard was denied a penalty after being brought down in the area early on, and Mellor hit the crossbar after being wrongly called offside.
But the possession counted late in the first half when Gerrard put Xabi Alonso through to put the Reds ahead.
Arsenal equalised against the run of play at the start of the second half, but Liverpool kept pushing for a winner.
Antonio Nunez made his belated debut, coming on for Florent Sinama Pongolle, and looked lively.
When it looked like Arsenal were going to deny Liverpool a win, up popped Neil Mellor in the final minute with a stunning strike beyond Lehmann.