If you download the mitab file corresponding to your organism (or the BIOGRID-ALL mitab if you have genes from multiple organisms), you'll see that the first two columns contain the Entrez gene IDs of the interaction partners, columns 3 and 4 contain the gene symbols. So you could use a short perl script like this (untested):

  open (FH,"<",$BIOGRID_file) or die ...
  while (my $line=<FH>) {
    chomp($line);
    my @tmpary = split(/\t/,$line);
    my @G1s = split(/\|/,$tmpary[2]); # gene symbol of partner 1
    my ($g1) = $G1s[0]=~/:(.+)/;
    my @G2s = split(/\|/,$tmpary[3]); # gene symbol of partner 2
    my ($g2) = $G2s[0]=~/:(.+)/;
    if ($wanted{$g1} && $wanted{$g2}) {
      # Assumes the symbols of the 5000 genes are keys to %wanted
      print "$g1\t$g2\n";
    }
  }
  close FH;