First, there is our 6th Article of Faith:
We believe in the same organization that existed in the Primitive Church, namely, apostles, prophets,pastors, teachers, evangelists, and so forth.
What the name of the Church should be was talked of in 3 Nephi 27:7-9:
The name of the Church in our time was given in Doctrine & Covenants 115:4:
For thus shall my church be called in the last days, even The Church of Jesus Christ of Latter-daySaints.
In our day many have believed that "Mormons" are presumptuous to call themselves Saints. For one (as seen in the previous scripture), it is a part of our name. For another, it is how we are live. The Bible Dictionary (in the LDS edition of the scriptures) defines Saint as:
The word saint is a translation of a Greek word also rendered “holy,” the fundamental idea being that of consecration or separation for a sacred purpose; but since what was set apart for God must be without blemish, the word came to mean “free from blemish,” whether physical or moral. In the New Testament the saints are all those who by baptism have entered into the Christian covenant
When we are baptized in the Church, we are accepting the Savior and His Gospel, and covenanting with Him. In turn, we are washed clean, and thus we start over again, free from blemish. So, it isn't that as Saints we're saying we are free from blemish. But after our baptism, we are striving to keep our lives free from blemish, which often requires repentance and accessing the Atonement to bring our lives back to that point.
And it isn't just us now. They were called saints in the former Church, as seen in Ephesians 2:19-20: