The Brazilian struck twice in the first half to put the visitors on course for their second win of the competition, with Pape Diouf's reply coming too late for Esbjerg.
Salzburg preserve their 100 per cent record in the pool while the Danes were unable to build on their 2-1 victory at Standard Liege last time out.
Salzburg, who hammered Elfsborg 4-0 in their opener, needed just six minutes to take the lead in Denmark as Rodnei combined with Brazilian compatriot Alan to supply the cross which the latter stabbed home left-footed.
Esbjerg responded with a strike from Mick van Buren and a header from Hans Henrik Andreasen, both of which were comfortably taken by Peter Gulacsi in the away goal.
Home goalkeeper Frederik Ronnow was also called into action to deal with a testing drive from Martin Hinteregger, but the Esbjerg defence was breached again after 38 minutes with Alan once more prodding home after Sadio Mane's assist.
The Austrian club could have been three goals up at the interval but for a wonderful save from Ronnow, who dived late to push away Kevin Kampl's powerful strike from distance.
Esbjerg showed more creativity after the break with Andreasen putting another header wide of the left post.
Van Buren then fired over the bar while Jonas Knudsen's strike from distance failed to hit the target as chances continued to go begging for the Danish hosts.
A goal finally arrived a minute from time with the help of Knudsen, who whipped a lovely ball into the area for Diouf to add the simplest of finishes.
However, an equaliser failed to follow and Jacob Ankersen's shot over the bar four minutes into stoppage time was as close as they came.