In a sudden death time control, if your clock runs out first, you lose. It doesn't matter if your opponent only had one second left, or less material, as long as he still had enough material to "theoretically" checkmate. (If he lacked such material, your timeout would have resulted in a draw.) This is true here as well as otb tournaments.
How do you win live chess games?

the fast games are difficult ... you have to balance moving quickly with playing well.
when i play i watch the clock too and try to make sure i have more time than my opponent. i also try to have a few options for moves ready ... i'm not good at live chess, but it hurts being beaten by a clock, especially if you're winning! :)

Live chess has always been a prob. Now what?
Well, first thing's first. Never give up. I understand that you timed out, so the problem is reasonably easy to solve. Start slow and steady.
1) Practice on Live Chess more often than Online Chess. - this way you can train yourself to play faster than online chess.
2) Set the time limit/timer a little lower after each sufficient number of games you have played. -eg. I set my first 6 games for 1 hour and after that set my next set of games for 45 minutes (but ofcourse, it depends on what timings and number of games you prefer).
3) Practice, practice and practice!
4) In the meantime, you can also work on your chess strategy as thinking to much might be taking up time. So, you can do puzzles and read books to maybe improve the way you think about a certain position and save plenty of time!
5) You will gradually improve!
Remember, this takes quite some time, effort and hardwork and there is a 100% gurantee you will improve if you follow these steps!
Hope this helps,
rockinz
i know this sounds like a daft question but how do you actually win "live chess" ? Like, i just played a game with someone it timed out on my turn. he only had 1.0 seconds left on the clock too. i had a queen, horse & 2 pawns, him, a rook and 2 pawns. Yet he won.... I tried to look around the faqs on how to win but no joy. Can someone help. thanks