There are a couple of ways you can improve on this. First, you can access the last element of an array by using the index -1. I.e.

$sort[scalar @sort - 1] == $sort[-1];
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Also, you can swap the values of two variables without a temporary variable in the middle:

($a, $b) = ($b, $a);  # swap the values of $a and $b
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So your code can be rewritten like so:

#!/usr/bin/perl -w
use strict;
use Data::Dumper;

my %hash = (Matt => 24,
            Dana => 19,
            Eric => 28,
            Sara => 20,
            John => 17,
            Mike => 23,);

print Dumper \%hash;

my @sort = sort {$hash{$a} <=> $hash{$b}} keys %hash;
($hash{$sort[0]}, $hash{$sort[-1]})
     = ($hash{$sort[-1]}, $hash{$sort[0]});

print Dumper \%hash;
[download]

-Matt