The Oracle 11.2 client package of drivers is installed on the same desktop PC and these drivers successfully connect my Toad client to the Oracle database.

unlikely story :) either you're missing %ORACLE_HOME% in your env

or the DBD/Oracle.dll is linked against something not compatible with what you have in %PATH% ( and %ORACLE_HOME%

To figure it out you could use depends.exe /c /f:1 /pb /ot:temp.txt ... perl.exe foo.pl

or open DBD/Oracle.dll in depends.exe