I solved this by removing the global identifier. !!!!!!!!!!!!

No, you solved it by making the incomplete statement
    $function_call = "memcpy(a, b, sizeof(a));"
(no ; terminates the statement) into a complete statement (by adding a ; and then not mentioning it to anyone, possibly even including yourself ;).

c:\@Work\Perl>perl -wMstrict -le
"my $function_used = 'memcpy';
 my $function_call = 'memcpy(a, b, sizeof(a));';
 $function_call =~ s/$function_used//g;
 print qq{'$function_call'};
"
'(a, b, sizeof(a));'
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(I'm assuming that by 'problem', you mean the fact that the OPed code will not compile.)

Update: If there's any chance that  $function_used may contain regex metacharacters, it's a very good idea to use quotemeta on it at some point, e.g.:
    $function_call =~ s/\Q$function_used\E//g;