DBD::Oracle uses a TNS connect string to connect to the database. Depending on your Oracle client install the TNS names could be stored in a flat file called tnsnames.ora, on a names server, or in OID (Oracle internet directory) repository. Anyway, define a TNS name which will connect to the database in question and test the connection with SQL*Plus. Once you are able to connect with SQL*Plus, try connecting as below:

$dbh = DBI->connect('dbi:Oracle:'.$TNS, $User, $Pass) || die $!;
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Update:
PLEASE take this with a grain of salt, as I only use TCP for connections to my databases. I played around with using the bequeath adaptor to connect to one of my test instances, and I added the following entry to the local tnsnames.ora file:

beqtst.world =
  (DESCRIPTION =
    (ADDRESS_LIST =
      (ADDRESS =
        (PROTOCOL = BEQ)
        (PROGRAM = oracle)
        (ARGV0 = oraclenboprd00)
        (ARGS = '(DESCRIPTION=(LOCAL=YES)(ADDRESS=(PROTOCOL=BEQ)))')
      )
    )
    (CONNECT_DATA = (SID = nboprd00))
  )
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The ORACLE_SID of the database to which I was connecting is "nboprd00". To use this TNS name with the connection code above, set $TNS='beqtst';

One other CYA, I did not test this with DBD::Oracle... I only tested it with SQL*Plus (I do not have DBD::Oracle installed on this test system, and I am not about to play with my install an hour before I leave for holiday. *Grins*)