"'When I came to the spring today, I said, "Lord, God of my master Abraham, if you will, please grant success to the journey on which I have come. See, I am standing beside this spring. If a young woman comes out to draw water and I say to her, 'Please let me drink a little water from your jar,' and if she says to me, 'Drink, and I'll draw water for your camels too,' let her be the one the Lord has chosen for my master's son."'" NIV translation
Eliezer explains that he had asked the Lord for a sign to show him God's will. We also note that he did so and was not even claiming God as his god but was doing so on the basis of Abraham's relationship with God.
"'Before I finished praying in my heart, Rebekah came out, with her jar on her shoulder. She went down to the spring and drew water, and I said to her, "Please give me a drink." She quickly lowered her jar from her shoulder and said, "Drink, and I'll water your camels too." So I drank, and she watered the camels also.'" NIV translation
The servant describes how Rebekah had fulfilled the sign that he had prayed for. We notice that it went down exactly as he had prayed.
"'I asked her, "Whose daughter are you?" She said, "The daughter of Bethuel son of Nahor, whom Milkah bore to him." Then I put the ring in her nose and the bracelets on her arms, and I bowed down and worshiped the Lord. I praised the Lord, the God of my master Abraham, who had led me on the right road to get the granddaughter of my master's brother for his son. Now if you will show kindness and faithfulness to my master, tell me; and if not, tell me, so I may know which way to turn.'" NIV translation
Eliezer explained how he had proposed to Rebekah on behalf of Isaac. We also see that he knows that he has done his part and that the decision is in their hands. That reminds us of the fact that God gives all men "free will" to make decisions.
"Laban and Bethuel answered, 'This is from the Lord; we can say nothing to you one way or the other. Here is Rebekah; take her and go, and let her become the wife of your master's son, as the Lord has directed.'" NIV translation
Now, we see that Rebekah's father made the decision based on the fact that he could tell that it was from God. It is interesting to note that they were in pagans in a pagan land and yet could see the hand of the one true God at work.