PDF::API2 confusingly has two text methods, one on PDF::API2::Page to get the text content, then a second on that PDF::API2::Content object to output the text. It's the second that takes the -underline => 'auto' option.

use strict;
use warnings;
use PDF::API2;

my $font_size=10;
my $pdf  = PDF::API2->new();
#A4 Landscap                                                          
+                                                                    
$pdf->mediabox('Letter');
my $page = $pdf->page;
# font settings                                                       
+                                                                    
my $fnt = $pdf->corefont('Helvetica',-encode => 'latin1');
my $boldfont=$pdf->corefont('Helvetica-Bold',-encode => 'latin1');

my $txt = $page->text();

my $top = 700;
my $left_margin = 50;

$txt->textstart;
$txt->font($boldfont,$font_size);

$txt->translate( 400, $top-65 );
$txt->text( "Trying to underline this sentence!",
            -underline => 'auto' );

$pdf->saveas('test.pdf');
[download]

Thank you, great help!