"It's a great day for the club and all the fans," the 19-year-old Gosling said. "I was thinking 'just shoot and get it on target.' We deserved to win, pressured them and played a lot of good football, and got the goal we deserved."
The substitute midfielder, who celebrated his birthday on Monday, broke the stalemate in the 118th minute when he collected an Andy Van Der Meyde cross just outside the Liverpool box and floated a ball into the top right of goal as defenders scrambled to close him down.
The meeting between the two Merseyside neighbors was the third in 17 days, with 1-1 draws in the Premier League and then in the cup, and at times it looked like it was going to be impossible to prize them apart until Gosling pounced to prevent a penalty shootout.
Everton will meet Aston Villa or Doncaster for a place in the last eight of football's oldest and most famous domestic cup competition.
It was resolute, uninspiring stuff from the opening whistle, big on heart but little on style. But it could all have been so different for Liverpool if Spain midfielder Xabi Alonso had scored in the 7th minute with a speculative drive that whistled past goalkeeper Tim Howard's post.
The Reds, buoyed by the weekend win against Chelsea, were beginning to control the game, but after 15 minutes talismanic captain Steven Gerrard limped off with a slight hamstring strain and was replaced by Yossi Benayoun.
Liverpool manager Rafael Benitez reorganized as Everton's hostile supporters serenaded Gerrard from the field, but The Toffees could still make little headway against a resolute Liverpool defense.
After 23 minutes, good work from Spanish midfield wizard Mikel Arteta, who was again at the heart of Everton's best moves, spread the ball to Leon Osman, onward to Tony Hibbert, who floated in a cross from the right that Liverpool goalkeeper Reina just palmed away.
Everton began to lift the tempo of a game that was at times as cold and uninspiring as its bleak 1970s designed stadium and Osman should have done better with a half-volley in the Liverpool box.
Everton defender Leighton Baines was easily marshaling an out-of sorts Fernando Torres, playing down the right channel, but as the game continued without a score, no substitution from either manager would appear to lift the gloom.
Liverpool, however, was getting stronger, with Alonso more frequently sending Albert Reina through in the inside left channel, but the final ball was charged down.
Everton substitute Jack Rodwell worked space in the 68th but could drill the ball only tamely at Reina, and in the 73rd Cahill, playing his 150th game for Everton, flicked the ball on from just inside the box but Osman, under pressure, could only strike the post.
Before this game the 22 players sent off in this fixture were a record in English football, but none had been in the FA Cup.
That all changed when Liverpool's Lucas Leiva received his second card of the game in the 76th when he brought down Joleon Lescott as the defender scurried past him with the ball in the center circle.
