Hi ukhare,
There is a number of things to point out in your code. Of course some of them are good practice while others are style and preference.

1. You should also use warnings; in your codes
2. As much as possible, try not to hard-core your file path in your program for portability issues. You might have to use the code some other places or some other time
3. use open 3 arguments options. Instead of 2
4. if you are using csv file, use Text::CSV or Text::CSV_XS module, instead of splitting or handpicking your data yourself.

Now, to your data, just like hdb said you weren't sorting. Something like so should give what you expected, though you didn't show us that.

use warnings;
use strict;

print sort grep $_ => <DATA>;

__DATA__
2016-02-02:00:44,mttsmshub1

2016-02-05:00:39,mttsmshub1

2016-02-03:00:32,tttsmshub1

2016-02-04:00:24,mttsmshub1
[download]

Assumed Output:

2016-02-02:00:44,mttsmshub1
2016-02-03:00:32,tttsmshub1
2016-02-04:00:24,mttsmshub1
2016-02-05:00:39,mttsmshub1
[download]

Hope this helps.

update:

In the solution I have provided earlier, there is no need for the function grep, the code would still have worked without it. Thanks AnomalousMonk for pointing that out. ++ for you!! There