Use grep to extract only the elements you require.

#!/usr/bin/env perl 
use strict;
use warnings;
use Data::Dumper;

my @array = (
    ['X1','X2','X3','X4' ],
    ['','foo','',''],
    ['bar','','','']
);

my @nofirstempty = grep { length $_->[0] } @array;
print Dumper \@nofirstempty;
[download]

  DB<14> @array = (['X1','X2','X3','X4' ], ,['','key', ''], ['key', ''
+, '']);

  DB<15> x \@array
0  ARRAY(0x600501580)
   0  ARRAY(0x6005d2818)
      0  'X1'
      1  'X2'
      2  'X3'
      3  'X4'
   1  ARRAY(0x6005015f8)
      0  ''
      1  'key'
      2  ''
   2  ARRAY(0x6004fa8c0)
      0  'key'
      1  ''
      2  ''

  DB<16>  for my $subarrayref ( @array ) {  $subarrayref = [ grep leng
+th $_,  @$subarrayref] }

  DB<17> x \@array
0  ARRAY(0x600501580)
   0  ARRAY(0x6005d2d70)
      0  'X1'
      1  'X2'
      2  'X3'
      3  'X4'
   1  ARRAY(0x600500e60)
      0  'key'
   2  ARRAY(0x600635c28)
      0  'key'
[download]

#!/usr/bin/env perl 
use strict;
use warnings;
use Data::Dumper;

my @array = (
    ['X1','X2','X3','X4' ],
    ['','foo','',''],
    ['','bar','',''],
    ['',rat','',''],
    ['me','','','']
);

##Some processing on @array
@array=???

my $qt = HTML::QuickTable->new();
$qt->render(\@array);

Output should be:
X1 X2 X3 X4
me foo
   bar
   rat

Currently it is:
X1 X2 X3 X4
   foo
   bar
   rat
me
[download]

That's a bit more involved but Array::Transpose softens the blow a bit.

#!/usr/bin/env perl 
use strict;
use warnings;
use Data::Dumper;
use Array::Transpose;

my @array = (
    ['X1','X2','X3','X4' ],
    ['','foo','',''],
    ['','bar','',''],
    ['','rat','',''],
    ['me','','','']
);

my @at = transpose (\@array);
my @condensed;
for my $row (@at) {
    my @shortrow = sort { (length ($b) > 0  <=> length ($a) > 0) } @$r
+ow;
    push @condensed, \@shortrow;
}
@array = transpose (\@condensed);
print Dumper \@array;
[download]

Removal of trailing rows just containing blanks is left as an exercise to the reader.

#!/usr/bin/perl

# http://perlmonks.org/?node_id=1195332

use strict;
use warnings;

my @array = (
    ['X1','X2','X3','X4' ],
    ['','foo','',''],
    ['','bar','',''],
    ['','rat','',''],
    ['me','','','']
);

use Data::Dump 'pp';
print "before\n";
pp \@array;

my @newarray;
my @itemsincolumn;

my $width = @{ $array[0] };

for my $row ( @array )
  {
  for my $column ( 0 .. $width - 1 )
    {
    length $row->[$column] and
      $newarray[ $itemsincolumn[$column]++ ][$column] = $row->[$column
+];
    }
  }
for my $row ( @newarray )
  {
  $_ .= '' for $row->@[0 .. $width - 1];
  }

print "after\n";
pp \@newarray
[download]

Sounds more like an issue with the output engine than with your data structure. For an alternative, you might like to check out Spreadsheet::HTML by jeffa.