Crespo's marvellous second-half effort was enough to steal the points for Claudio Ranieri's side and separate these two west London rivals.
Fulham started brighter and created the first opening of the game just two minutes in following some crafty play by Sean Davis.
The midfielder found Lee Clark on the wing and his left-foot cross was met by the head of Louis Saha and only a great save by Carlo Cudicini denied the striker.
Three minutes later Chelsea came close courtesy of Irishman Damien Duff. The Blues midfielder cut inside before firing a 25-yard right-foot shot which Edwin van der Sar did well to palm away.
It turned out to be Duff's last contribution of the game as he landed awkwardly following a challenge from Davis and Joe Cole was brought on in his place.
On 13 minutes Crespo was put through on goal after Adrian Mutu charged down a Zat Knight clearance.
The Argentinian managed to get a shot away despite claims of a pull back by the trailing Andy Melville but his effort rolled wide.
Chelsea turned up the pressure on their local rivals as Crespo and Mutu both went close, but it was defender Glen Johnson who almost broke the deadlock on 30 minutes.
A Mutu corner caused chaos in the Fulham box and the ball fell to Johnson 15 yards out. The full-back hit a shot which seemed destined for the top corner - only for Jon Harley to divert the ball over the bar with his head.
The visitors went close again with a corner as Terry met Crespo's flick on, but Van der Sar was equal to the effort as he stuck out a leg to keep the ball out.
Davis limped off eight minutes before the break and on the verge of half-time Fulham came within inches of breaking the deadlock.
Steed Malbranque's inswinging free-kick was met by the head of captain Melville - but his effort was straight at the Chelsea keeper.
Seven minutes into the second period Frank Lampard was gifted a great chance. Crespo did well to pick out the England international, but Lampard's finish was poor as his sidefooted effort was easily saved.
Saha appealed for a penalty two minutes later when he appeared to be pulled down by Terry - but it was on 62 minutes that the opening goal finally arrived.
Wayne Bridge fired in a great cross and Crespo rose ahead of Fulham defender Melville to steer a superb header into the top corner via Van der Sar's left-hand post.
Luis Boa Morte almost pulled the Cottagers level moments later as Cudicini managed to save a fierce drive with his legs.
Barry Hayles was thrown into the picture in a bid to rescue a point but Fulham still failed to create any real chances.
Clark fired over the bar and Malbranque had an effort beaten away, while Sylvain Legwinski hit a long-range effort marginally over the bar ten minutes from time.
Lampard could have sealed the game minutes later but he was denied by a great stop by van der Sar.