Klasnic scored the only goal in Wednesday's fourth-round replay against Wigan and was on target again at Craven Cottage with a volley from 12 yards.
Bolton have not graced the FA Cup final since the late, great Nat Lofthouse scored twice to beat Manchester United in 1958. They have now taken two significant steps in the space of a week.
This victory was not without its controversies. Tempers boiled over in the second half with an off-the-ball half between Chris Baird and Martin Petrov, the latter catching Baird with a flailing arm and the Fulham defender responding by shoving the Bolton winger in the face. Both men were fortunate to only be booked.
Bolton's physical centre-back pairing of David Wheater and Gary Cahill were effective in crowding out Andrew Johnson and Moussa Dembele, if not always legal.
Fulham were at least able to cheer the return to action of striker Bobby Zamora, who came off the bench after five months out with a broken leg.
Mark Hughes made one change to his starting line-up from Monday's Premier League draw with Chelsea, with Zoltan Gera replacing the injured Damien Duff in midfield.
Bolton brought back striker Kevin Davies and midfielder Stuart Holden following their midweek cup win, while Marcos Alonso started at left-back in place of the injured Sam Ricketts.
Fulham started brightly. Clint Dempsey beat Cahill to flick the ball into the path of Dembele, whose shot was deflected away by Bolton keeper Adam Bogdan.
Holden had Bolton's first shot on goal as he latched onto a knock-down from Davies on the edge of the Fulham box but his left-footed effort flew wide.
Dempsey was dominating the middle of the park and he pounced on a loose piece of control from Alonso and released Gera down the right, who whipped in a teasing low cross that was just too far ahead of Johnson.
But it was Bolton who took the lead after 18 minutes with a left-foot volley from Klasnic after some clever approach play from Chung-yong Lee.
The South Korean completed a neat one-two with Klasnic on the edge of the Fulham box before Fabrice Muamba drilled in a cross that Aaron Hughes failed to deal with cleanly.
As the ball dropped inside around the penalty spot, Klasnic was the first to react and he rifled his shot past Mark Schwarzer.
Fulham continued to press and Dembele's misplaced pass ended up creating a shooting opportunity for Gera but the Hungarian fired over the top.
Wheater was fortunate not to receive only a lecture from referee Stuart Attwell after he hauled Dembele to the ground in the build-up to a promising move.
Johnson was lively up front for Fulham, trying desperately to steal a yard in between the two centre-backs but with little reward.
Cahill went through the back of Johnson once but the new England international also pulled off a perfectly-timed interception on the edge of his own box.
Fulham were having to battle against Bolton's physicality but they did have chances, with Brede Hangeland playing Dempsey into space 25 yards out only for the American to drag his low shot wide of the near post.
Fulham made a lively start to the second half with Dembele lifting one shot over before forcing Bogdan into a smart save wiht a low curling effort from the edge of the box.
Muamba was booked by Attwell when he halted Dembele's run by scything him down from behind. Tempers were beginning to fray, with Petrov and Baird clashing off the ball.
Sidwell cut in from the left and tested Bogdan from distance while, at the other end, Baird denied Davies a shot on goal with a well-timed challenge in the box.
Schwarzer kept Fulham in the game by clawing away Klasnic's close range effort after the Bolton striker had been picked out in the box by Lee.
Fulham won a free-kick 25 yards from goal after Davies was judged to have fouled |Danny Murphy. Robinson charged furiously to protest and was booked for dissent.
Murphy tried to bend his shot around the wall but it floated over the far post.
Petrov then found himself in acres of space in the Fulham box but he took one shot too many and his fierce drive was tipped around the near post by Schwarzer.
Fulham threw the kitchen sink at Bolton in the closing stages but the Trotters clung on to seal the victory, with Cahill producing a magnificent block from Dempsey's shot inside the box.