Update: I now realize I was unintentionally redundant. I opened the reply form, then got distracted; by the time I submitted, someone else had come up with essentially the same concept. My apologies! On the upside, that does prove that it's a good idea. :)

Do this, maybe:

while <$FILE_1> {
   $file1{$_}=0;
}

while <$FILE_2> {
   my ($match_val) = split(/\s+/, $_); #split on whitespace
   print $_ if defined $file1{$match_val};
}
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Searching hash keys is faster than a linear array search (especially for large constructs). The first loop loads a hash where the keys are the data from file 1 (the values don't matter here). The second loop prints each line in file2 that has a value in its first column that matches a hash key.

Should be pretty fast, and has the added advantage of not reading all of file 2 into memory.