The main trick is to double the number of columns plus one and to fill the void symmetrically...

use strict;

sub pascalMatrix {
  my $tabularRows = "";
  my @row = (1);
  my $cols = ($_[0] * 2) + 1;   #add: final number of columns
  my $filler = "{ }";           #add: used to mark an empty entry

  for (my $r = 0; $r <= $_[0]; $r++) {
    # Output the current row

    ### begin of modification
    #  insert a filler between each item (sprintf is for LaTeX cosmeti
+cs)
    my @fmt = map { (sprintf("%3d",$_), $filler) } @row; pop @fmt;
    #  pre-/append a filler until final number of columns reached
    unshift(@fmt,$filler), push(@fmt,$filler) while (@fmt < $cols);
    $tabularRows .= join (" & ", @fmt) ." \\\\\n";
    ### end of modification


    # Generate the next row
    my @nextRow = (1);
    for (my $c = 1; $c <= $r; $c++) {
      push @nextRow,
    @row[$c - 1] + @row[$c];
    }
    push @nextRow, 1;
    @row = @nextRow;
  }
  return
    "\\begin{tabular}{*{$cols}{c}c}\n" .   # changed!
      $tabularRows .
    "\\end{tabular}\n";
}

print pascalMatrix(10);
[download]

This was the output of the original version:

\begin{tabular}{*{5}{c}c}
1 \\
1&1 \\
1&2&1 \\
1&3&3&1 \\
1&4&6&4&1 \\
1&5&10&10&5&1 \\
\end{tabular}
[download]

This is the output of the patched version:

\begin{tabular}{*{11}{c}c}
{ } & { } & { } & { } & { } &   1 & { } & { } & { } & { } & { } \\
{ } & { } & { } & { } &   1 & { } &   1 & { } & { } & { } & { } \\
{ } & { } & { } &   1 & { } &   2 & { } &   1 & { } & { } & { } \\
{ } & { } &   1 & { } &   3 & { } &   3 & { } &   1 & { } & { } \\
{ } &   1 & { } &   4 & { } &   6 & { } &   4 & { } &   1 & { } \\
  1 & { } &   5 & { } &  10 & { } &  10 & { } &   5 & { } &   1 \\
\end{tabular}
[download]

My LaTeX installation is gone, but there's a nice LaTeX Online-Compiler available.
HTH

Update: The code presented above is a runnable Perl program. In order to make it work with LaTeX you need to get rid of use strict; and print pascalMatrix(10);. Furthermore, replace sub pascalMatrix by \perlnewcommand{\pascalMatrix}[1]. Remove, rename, or uncomment the previous version of \pascalMatrix.