But it didn't tell the full story as the Premier League visitors rode their luck before earning a replay.
Mark Gower spurned two great chances and Nathan Dyer was inches away from causing an upset and sending the Londoners crashing out in the fifth round.
Roy Hodgson made three changes from the side that played out a goalless draw against Wigan last time out.
But his side were handed a massive scare inside three minutes when Gower hit the post.
A neat one-two with Scotland saw the Swansea winger beat goalkeeper Mark Schwarzer only for the ball to bounce off the post and away to safety.
Another sweeping move should have then seen Gower hand Swansea a 23rd minute lead.
Jordi Gomez and Rangel combined for the ex-Southend winger to fire straight at Schwarzer from close range with the goal at his mercy.
But Swansea were made to pay on the stroke of halt-time after Monk's own goal.
Paul Konchesky's floated corner was missed by goalkeeper Dorus de Vries and the loose ball bounced off the home captain's hip for Fulham's opener.
Gower's luck showed no sign of ending as his shot was pushed onto the crossbar by Schwarzer.
But Swansea wouldn't be denied as Scotland twisted past Aaron Hughes before expertly firing home in the 52nd minute for his 17th goal of the season.
It was the least the home side deserved as a Fulham side, who had yet to win on their travels in the Premier League, were pegged back for long periods.
Gomez came close before Dyer hit the side-netting in the closing stages as Fulham clung on to earn a replay at Craven Cottage.