Here is a start..

    Export["test.txt", "a1,b1,c1,d1,\"e1\ne1\ne1\"\na2,b2,\"c2\nc2\nc2\",d2,e2"]

    i = 0;
    StringSplit[StringJoin[#], ","] & /@ 
      Split[Characters[Import["test.txt", "Text"]],
       (If[# == "\"", ++i]; ! (# == "\n" && EvenQ[i]) ) &]

Note this will break if there are commas within quoted strings.

Edit: a bit more elegant..and this takes care of the comma-within-quotes issue.

    stringsplitnoquote[string_String, d_String /; StringLength[d] == 1] :=
      Module[{i = 1},
      StringSplit[
       string, _?((If[# == "\"", i *= -1]; i == 1 && # == d) &)]]

    stringsplitnoquote[#, ","] & /@ 
         stringsplitnoquote[ Import["test.txt", "Text"] , "\n"]


Due credit.. borrowed from here : http://mathematica.stackexchange.com/a/103030/2079