First-half goals from skipper Jiloan Hamad and Magnus Eriksson, just two minutes apart, ensured victory for the Swedish league leaders and left Hibs facing an uphill task in the return leg next Thursday.
Midway through their domestic season, Malmo's extra sharpness was clear throughout, but it was Hibernian who carved out the first chance of the night when after just seven minutes when Owain Tudur Jones rose unmarked to meet Craig's free-kick - but the summer recruit from Inverness Caledonian Thistle headed wastefully over.
And within three minutes, the Scots were behind. Skipper James McPake was booked for an apparent tug on striker Tokelo Rantie as the South African international looked to race onto a through-ball and Hamad sent his 20-yard free-kick into the net via the underside of the crossbar.
Within two minutes, the costly start was complete as Malmo doubled their lead.
Emil Forsberg created space on the left side of the penalty box and his cutback was slotted home by Eriksson.
Hibs plugged away and Rowan Vine watched as his deflected drive agonisingly rebounded off the post with goalkeeper Johan Dahlin helpless.
However, the Scottish Premier League side had goalkeeper Ben Williams to thank for ensuring things did not get even worse when his outstretched boot diverted Eriksson's shot over for a corner, which resulted in defender Erik Johnsson heading wide.
Williams saved from Rantie and from Miiko Albornoz and Hamad as there was little respite for Hibs at the start of the second period.
And the visitors were fortunate to survive another sweeping attack as Simon Thern's shot screwed across goal off the inside of Williams' left-hand post.
In the end, after repelling everything Malmo could throw at them in the second half, Hibs will have been delighted to get to the full-time whistle with just a two-goal deficit to chase next week back in Scotland.