There isn't a verse like that but there are verses like this:
1 Tim 4:16 Take heed unto thyself, and unto the doctrine; continue in them: for in doing this thou shalt both save thyself, and them that hear thee.
1 Tim 4:16 Take heed unto thyself, and unto the doctrine; continue in them: for in doing this thou shalt both save thyself, and them that hear thee.
If you fell into a well with no way out and someone threw down and rope and pulled you out, "in a sense" you can say that you "saved yourself" by choosing to grab and hold onto the rope until that person pulled you out, but ultimately, the person who pulled you out of the well saved you and ultimately, it is Jesus Christ who saves us.