After victories over Faroe Islands minnows NSI Runavik and Northern Ireland part-timers Crusaders, the Cottagers had been expected to secure another win at Dugopolje’s Stadion Hrvatski Vitezovi.
However, Split proved dogged opponents throughout and were unfortunate not to take the lead in just the sixth minute.
The Reds peppered Mark Schwarzer’s goal with numerous chances and Ante Erceg came closest, although he somehow shot wide when through on goal.
The home side missed another gilt-edged chance to score in the dying moments, with substitute Duje Cop missing an equally inviting chance after turning his marker.
Fulham had chances of their own but struggled to break down the resilient Croatian side, to the disappointment of 86 travelling fans.
Heading into the match Martin Jol named a strong starting line-up, with Andrew Johnson and Bobby Zamora again given a chance to hone their strike partnership.
New signings Pajtim Kasami, Csaba Somogyi and Marcel Gecov were all named amongst the Cottagers’ substitutes, while the home side included former Zenit St Petersburg defender and 2008 UEFA Cup winner Ivica Krizanac in their starting line-up
Split supporters welcomed Fulham onto the pitch with a sign reading ‘resistance is futile’ and the visitors were almost forced to heed that warning in the opening minutes of the match.
Ante Erceg fired a shot across goal in the fourth minute before the same player was put through on goal, but the 21-year-old screwed his shot wide when one-on-one with Mark Schwarzer.
The attempt raised the volume in Dugopolje as the visitors struggled to string passes together, with a blocked Damien Duff shot the Cottagers’ only moment of note in the opening 25 minutes.
Fulham were continually exposed down the flanks and their frustrations were evident when Dickson Etuhu was booked in the 21st minute for clattering Krizanac.
Split came close from the resulting free-kick, which was driven into the box for Bojan Golubovic to hook over Schwarzer – only for the Australian to backpedal and save.
Split goalkeeper Andrija Vukovic was forced into his first save of the match on the half-hour mark when Zamora fired in an ambitious 25-yard drive after some neat team play.
While the end product was lacking, the Cottagers were beginning to control possession and Johnson forced the goalkeeper into action once more eight minutes later from a tight angle.
RNK, though, were still proving dangerous and an ambitious 30-yard effort from Ante Vitaic looked dangerous until it took a deflection.
Fulham began the second period more positively and came close to scoring five minutes after the restart when Johnson broke down the right and cut the ball back for Matthew Briggs, whose shot into an empty goal was thwarted by a lunging challenge in the six-yard box.
John Arne Riise came close to finding the bottom corner with a low drive three minutes later as the visitors began to take more control of the game, before Zamora blazed a long-range effort wide.
Split began to press once more as the game entered the latter stages and Schwarzer was forced to save when Cop managed to turn his marker and force a shot in the 18-yard box.
Jol brought on Steve Sidwell for Etuhu in a bid to change the game but the only impact the former Aston Villa midfielder made was receiving a yellow card a minute after coming on.
The Fulham boss made another switch with 10 minutes remaining, handing recent signing Kasami his debut in place of the ineffective Zamora.
However, the former Palermo midfielder’s introduction failed to gee up his new team-mates and Cop twice came close to scoring in the final 10 minutes.
First he darted a header across the face of the goal, before having the home faithful on their feet when he neatly turned Chris Baird but could only manage a pitiful shot which Schwarzer easily stopped.