I think this is a common happening. I know I prefer to start out my plants in tiny containers (solo cups that I've perforated on the bottom and sides with a sewing needle to allow the soil to vent) so that I can replant them after a time and bury the stems to just under the tiny round baby leaves (cotyledons). It gives them extra support, and encourages them to grow more roots from the buried portion of the stem.
You can also try using a plastic drinking straw or a chopstick as a stake to prop up your plant, lean it against or gently tie it with a little fishing line or the like (dollar store nylons, cut into small strips, are soft, stretchy, and will not rot).