Since you only want this occur when the Password entry widget is active, you should bind the '<Return>' key to $pass_en (and note that you have to do it -after- button1 has been defined).  Try the following (noting that the line in blue is the crucial one):

use Tk;
my $main = MainWindow->new;
$main->configure(-title => 'Verify User', -background=>'blue');
$main->geometry('+300+300');
my $top = $main->Frame(-background=>'blue',
    -relief=>'sunken')->pack(-side=>'top',-fill=>'x');
my $left1 = $top->Frame(-background=>'blue',
    )->pack(-side=>'left',-pady=>9,-padx=>8);
my $left2 = $top->Frame(-background=>'blue',
    )->pack(-side=>'left',-pady=>9,-padx=>8);
my $user_frm = $left1->Frame(-relief => 'groove',
                    -borderwidth=>3, -background=>'blue',
                    )->pack(-side=>'top',-fill=>'x');
my $user_lbl = $user_frm->Label(-text=> 'Enter Username',
                        -background=>'blue',-foreground=>'white',
                        )->pack(-side=>'left');
my $user_en = $user_frm->Entry(-width=>8,
                       -background=>'white',)->pack(-side=>'left',-pady=>3);
my $pass_frm = $left1->Frame(-relief => 'groove',
                    -borderwidth=>3, -background=>'blue',
                    )->pack(-side=>'top',-fill=>'x');
my $pass_lbl = $pass_frm->Label(-text=> 'Enter Password',
                        -background=>'blue',-foreground=>'white',
                        )->pack(-side=>'left');
my $pass_en = $pass_frm->Entry(-width=>12,
                       -show=>'*',
                       -background=>'white',)->pack(-side=>'left',-pady=>3); 
my $button = $left2 ->Button(-text => 'Exit', -underline => 0,
                     -width => 10,
                     -borderwidth=>3,
                     -command => sub{do_exit()}
                     )->pack;
my $button1 = $left2 ->Button(-text => 'Connect ',
                     -width => 10,
                     -borderwidth=>3,
                     -command => sub{do_submit("$user_en, $pass_en")}
                     )->pack;

$pass_en->bind("<Return>", sub { $button1->invoke });


MainLoop;

sub do_submit() { print "Called 'do_submit'\n"; }
sub do_exit()   { print "Called 'do_exit'\n"; }

Note that I've supplied two stubs so that you'll have something to actually call when the buttons are invoked.   I've also added '-' to a few switches where it was missing, to get rid of warnings (eg. "pady" => "-pady").  Hope this helps!

liverpole is correct. But, to be complete, you also want to add a binding to $user_en

$user_en->bind("<Return>", sub { $button1->invoke });

Many thanks, that has worked a treat.

++ Very nice, I've been tinkering with Tk a little more lately and the more I romp around the more I learn. This is exactly what I needed $main->geometry('+300+300');