I doubt this counts as more elegant, but...

use strict;
use warnings;

my $text = q{
I need to extract the first several words from a paragraph of text con
+tained in a $var, 
so as to get the longest extract that's less than or equal to 200 char
+acters. 
My brute-force-and-ignorance method is: blah some more text here etc e
+tc
};

my $chunk;

($chunk) = $text =~ /^(.{1,200})(?<!\s)(?!\w)/;

printf "[%d] %s\n", length($chunk), $chunk;

$chunk = $text;

$chunk =~ s/^(.{200}).*$/$1/;
$chunk =~ s/^(.*)\s+\w*$/$1/;

printf "[%d] %s\n", length($chunk), $chunk;

$chunk = "";
$text =~ s/^\s+//;

foreach (split /\s+/, $text) {
    if (length($chunk) + length($_) <= 200) {
        $chunk .= $_ . " ";    
    } else {
        last;
    }
}

chop $chunk;
printf "[%d] %s\n", length($chunk), $chunk;
[download]

Output:

Use of uninitialized value in length at C:\temp\ws.pl line 15.
Use of uninitialized value in printf at C:\temp\ws.pl line 15.
[0] ''
[242] '
I need to extract the first several words from a paragraph of text con
+tained in
a $var,
so as to get the longest extract that's less than or equal to 200 char
+acters.
My brute-force-and-ignorance method is: blah some more text here etc e
+tc
'
[194] 'I need to extract the first several words from a paragraph of t
+ext contai
ned in a $var, so as to get the longest extract that's less than or eq
+ual to 200
 characters. My brute-force-and-ignorance'
[download]

gav^