In the New Testament, it's found in Matthew 28:18-20:
"Then Jesus came to them and said, “All authority in heaven and on
earth has been given to me. Therefore go and make disciples of all
nations, baptizing them in the name of the Father and of the Son and
of the Holy Spirit, and teaching them to obey everything I have
commanded you. And surely I am with you always, to the very end of the
age.”"
Just to add a bit more theology to it, according to this and other Systematic Theology (See Wayne Grudem's Systematic Theology, chapter 18 on Prayer), doing something 'In the name of Jesus', as we see so frequently through the book of Acts, is a reference to 'in the authority of Jesus'. Therefore, adding "In Jesus' name" at the end of the prayer is not the be all and end all - instead, it is, to quote Grudem:
"To come in the name of someone means that when another person has
authorized us to come on his authority, not on our own. When Peter
commands the lame man, 'in the name of Jesus Christ of Nazareth, walk'
(Acts 3:6), he is speaking on the authority of Jesus, not on his own
authority".