וָאֹמַ֖ר אֶל־אֲדֹנִ֑י אֻלַ֛י לֹא־תֵלֵ֥ךְ הָאִשָּׁ֖ה אַחֲרָֽי׃
Rashi notes that the word אֻלַ֛י here is written chaseir, without the vuv, unlike how it is written earlier in the parsha, so if you didn't know better you might read it as אֵלַי. The Sanz Klausenberger Rebbe explains that to be the shliach to find a wife for Yitzchak is not a simple undertaking. When Hashem came to Moshe and told him that he was to be the go'el of Klal Yisrael, Moshe responded by saying, "Mi anochi?" Who am I to take on a job like this? Choose someone more worthy. Here too, Eliezer worried that his mission might fail because he was not the right one for the job -- who says the right girl will come with ME, אֵלַי? Maybe someone else better suited would have more success. A role, a mission, a shlichus has to be undertaken with modesty and humility and cognizance of the responsibility involved. It is precisely that feeling of unworthiness which made Eliezer successful in completing his task.
We see throughout the parsha that Eliezer does his utmost to take himself out of the picture. Nowhere in the story is his name even mentioned -- he is just referred to as the "eved," Avraham's servent. He views himself as just an instrument of Avraham's, not a gavra in his own right who deserves the credit. He davens הַקְרֵה־נָ֥א לְפָנַ֖י הַיּ֑וֹם, that what happens should be a mikreh, like a chance occurrence (see R' Shimshon Pincus's hesber), not something that happens due to his planning, his input, his effort. He is willing to take a backseat and let Hashem take control and work things out.
This same idea is brought out in a diyuk of the Meshech Chochma. Just as Eliezer finishes davening that the right girl should come and offer him water, the Torah tells us וַֽיְהִי־ה֗וּא טֶ֘רֶם֮ כִּלָּ֣ה לְדַבֵּר֒ וְהִנֵּ֧ה רִבְקָ֣ה יֹצֵ֗את. Grammatically, the words of the pasuk seem out of order. We would expect it to say וַֽיְהִי טֶ֘רֶם֮ כִּלָּ֣ה ה֗וּא לְדַבֵּר֒. Why does the word ה֗וּא appear before the words טֶ֘רֶם֮ כִּלָּ֣ה? Meshech Chochma explains:
דלפי הנראה הבקשה של אליעזר הוא הסיבה ומה שרבקה יצאת וכדה על שכמה הוא המסובב. אולם האמת לא כן כי השם רצה להראות שרבקה בת זוגו של יצחק שם הדברים בפי אליעזר שיעשה נסיון בהשקיית מים וידע מזה שרבקה בת זוגו וא״כ יציאת רבקה הוא הסיבה והמסובב הוא שידבר זה אליעזר בזה שאמר ויהי מה הוא שהיה שהוא טרם כלה לדבר היינו שדבורו היה מהשם ודו״ק בזה.
וַֽיְהִי־ה֗וּא tells us that the situation that was, i.e. the whole test of drawing the water and Rivka's response, was already in place טֶ֘רֶם֮ כִּלָּ֣ה לְדַבֵּר֒ , before Eliezer prayed for it to happen. His tefilah was not the cause that set events in motion. Aderaba, it was the events that Hashem had set in motion which caused Eliezer's tefilah. Hashem put the words in Eliezer's mouth so that when Rivka would pass his test, it would be clear that she was indeed the one destined for Yitzchak.
Eliezer was just a kli b'yad Hashem, not that his choices, his actions, or even his prayers were the cause of anything.
And that's exactly how he wanted it to be.