The match began at a frantic pace, with Kenny Dalglish's Liverpool creating the first chance in the second minute. Andy Carroll cut the ball back for Jay Spearing on the edge of the area, but the midfielder curled his effort over the bar.
Everton's first major sight of goal arrived four minutes later when Leighton Baines found the roof of the net from distance.
As the tie began to settle down, Everton took a 23rd-minute lead thanks largely to an error from Liverpool defender Carragher. The veteran defender's attempted clearance struck Tim Cahill on the edge of the box. The ball then fell to Jelavic, who slotted past Brad Jones in the Liverpool goal to hand the Toffees the advantage.
In reply Carroll and Luis Suarez linked for the latter to send in a cross which captain Steven Gerrard was inches away from making connection with.
The Reds continued to dominate the possession for the remainder of the half, but they were unable to force an equalising goal.