I think the function you're looking for is "invoke", which causes the widget object (in this case, the Login button) to be pressed when 'Return' is entered.

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!