I would suggest using Graham Barr's time-tested and lean Date::Parse, somewhat like this:

use Date::Parse qw(strptime);
for (<DATA>) {
    my @date =  (strptime($_))[4,3,5];
    ++$date[0];
    $date[2] += 1900;
    print join('/' => @date) . $/;
}
[download]

which parses/outputs correct for your sample DATA above.
And if you insist on not using additional modules. I'd rather steel from its code ...