This sound more of an issue running SQL scripts in sqlplus than perl issue. So I would expect the EXIT command to be fed into SQLPLUS if it is not in the mySql.sql script.
my $sqlsource = 'mySql.sql';
open( SQLPLUS, "|sqlplus /nolog") or die "Can't run sqlplus\n" ;
print SQLPLUS <<SQL;
connect USERNAME/PASS\@INST
\@$sqlsource
EXIT
SQL
close SQLPLUS;