The 21-year-old showed great composure to open the scoring in the seventh minute when he pounced on a defensive error before driving his shot between the Randers goalkeeper's legs.
It's Mabil's fourth goal this season for Esbjerg to go with his four assists as he continues to impress in his loan spell from FC Midtjylland.
Robin Soeder doubled Esbjerg's lead in the 46th minute to secure the three points and lift the Danish side off the bottom of the table and into 12th position.
Fellow Australian Brent McGrath came off the bench for Esbjerg in the 90th minute.
Meanwhile, Aussies Mustafa Amini and Chris Ikonomidis both started for AGF in their 1-0 loss to FC Copenhagen, but both were substituted before full-time as AGF continues to slide down the table.
Elsewhere in Europe, Aussies Craig Goodwin and Kenneth Dougal's Sparta Rotterdam went down 1-0 to Vitesse, with Goodwin being substituted in the 69th minute.
Brad Jones kept another clean sheet, his eighth in the Eredivisie this season, as Feyenoord won 4-0 to stay top of the table.
Former Adelaide United midfielder Jimmy Jeggo started and received a yellow card in Stum Graz's 1-0 loss to Salzburg.
Robbie Kruse was left out of Bayer Leverkusen's squad as they defeated Schalke 1-0 in the German Bundesliga.