The Whites continued to press, although at the other end Molins made a more telling contribution, threading through to Daniel Larsson who was expertly beaten to the ball by Aaron Hughes.
The Northern Ireland defender also did well to block Edward Ofere moments later, with the Malmo striker taking up a dangerous position in the Fulham box.
Managed by former Sheffield Wednesday and Coventry defender Roland Nilsson, the hosts certainly suggested they were a side that would pose problems, with Brede Hangeland also on hand to out-jump Ofere and see the ball to safety.
Having found their early rhythm, it was Fulham that again went closest to a breakthrough, with Zamora meeting Bjorn Helge Riise's excellent delivery from the right. Unfortunately, last season's top goal-scorer's header dropped the wrong side of the bar.
With 20 minutes gone Murphy tested Johan Dahlin from distance, while Zamora had the ball in the net moments later, only to be ruled offside from Zoltan Gera's clever through ball.
An equally intelligent free-kick from Molins tested David Stockdale, with the young 'keeper quick to react to the midfielder's near post effort.
Impressing from the off, Dickson Etuhu could well have put his side in front with the goal at his mercy, but the Nigerian international took his eye off Zamora's teasing pull-back.
Malmo also saw a chance come and go on 35 minutes, with Miljan Mutavdzic seeing his strike deflect wide off the boot of Freddie Stoor when well placed.
The Whites certainly finished the half on top, in what was a controlled and professional 45 minutes of football.
A positive start to the second-half followed, with Zamora all but finding the breakthrough but for some brave goalkeeping from substitute Dusan Melicharek five minutes in.
Chris Baird almost found the net with a deflected strike from Gera's cross, while Hughes rose highest to one corner from the right.
Simon Davies fashioned a half chance for Zamora on the hour as Fulham continued to threaten.
That said, the home side still frustrated Fulham at every opportunity, showing exactly why they currently sit second in the Swedish Allsvenskan, five points behind leaders Helsingborgs.
Hughes was called into action again with 12 minutes remaining, beating Dardan Rexhepi to what looked a certain goal from Jiloan Hamad's drag back.
With the game drawing to a close Eddie Johnson flashed a low delivery across the box, where David Elm was unlucky not to turn the ball past the Malmo 'keeper.
Teams
Malmo: Dahlin; Vinzents, Pernennel, Andersson, Elanga; Molins, Mutavdzic, Figueredo, Durmaz; Ofere, Larsson
Fulham: Stockdale; Stoor (Kamara 76, Smith 79), Hangeland, Hughes, Konchesky; Duff (Davies 45), Etuhu (Baird 45), Murphy (Kelly 63), Riise (Elm 45); Gera (Greening 63), Zamora (E.Johnson 76).
Read more: http://www.fulhamfc.com/MatchAndTeam/MatchCentre/Matches/1011/PreSeason/MalmoAway.aspx#ixzz0uvPrtbPi