I had similar results even with longer arrays:

#!/usr/bin/perl
use strict;
use warnings;
use Benchmark qw( cmpthese );

my @array = map { int rand 100 } 1 .. 1_000_000;
cmpthese -10, {
   op => sub {
      my ($min, $max, $avg) = min_max_avg(\@array);
   },
   simple => sub {
      my ($min, $max, $avg) = min_max_avg_simple(\@array);
   },
};

sub min_max_avg_simple {
   my $ref = shift;
   
   my $n_elements = scalar @$ref;
   return unless $n_elements > 0;
   
   my ( $min, $max, $sum ) = ($ref->[0]) x 3;
   my $i = 1; # Start from next element
   while ($i < $n_elements) {
      my $value = $ref->[$i];
      if ($value < $min)
         { $min = $value }
      elsif ($value > $max)
         { $max = $value }
      $sum += $value;
      ++$i;
   }
   return ( $min, $max, $sum / $n_elements );
}

sub min_max_avg {
    my $ref = shift;
    my ( $min, $max, $agv, $i );
    my ( $current_min, $current_max );
    if ( @{$ref} % 2 == 0 ) {
        ( $min, $max ) =
          $ref->[0] < $ref->[1]
          ? ( $ref->[0], $ref->[1] )
          : ( $ref->[1], $ref->[0] );
        $agv = $ref->[0] + $ref->[1];
        $i   = 2;
    }
    else {
        $min = $max = $agv = $ref->[0];
        $i = 1;
    }
    while ( $i < @{$ref} ) {
        ( $current_min, $current_max ) =
          $ref->[$i] < $ref->[ $i + 1 ]
          ? ( $ref->[$i], $ref->[ $i + 1 ] )
          : ( $ref->[ $i + 1 ], $ref->[$i] );
        $min = $current_min if ( $current_min < $min );
        $max = $current_max if ( $current_max > $max );
        $agv += $ref->[$i] + $ref->[ $i + 1 ];
        $i   += 2;
    }
    return ( $min, $max, $agv / @{$ref} );
}

__END__

       s/iter     op simple
op       1.13     --   -37%
simple  0.707    60%     --
[download]

Next time I'll wait a bit more before coding :)