I have a more technical answer.
The Gemera in Megillah 22a says, כֹּל פְּסוּקָא דְּלָא פַּסְקֵיהּ מֹשֶׁה אֲנַן לָא פָּסְקִינַן לֵיהּ, “any verse that Moshe did not divide, we may not divide.” Meaning we can’t split pesukim wherever we want. So since in the Haggadah we want to say that we saw Hashem's great hand and since in that pasuk we can’t just stop wherever we want (Shemot 14:31), therefore we continue the pasuk, even though Moshe’s name is mentioned. (And the Haggadah uses this specific pasuk since it felt it represents Hashems greatness in the best way, even if there are other peskuim that say so).
You may ask, we have other half pesukim in the Haggadah! @Ploni here says, "the Tzitz Eliezer (9:17:10) cites the Sfat Emet asks how the Hagadah used on Pesach can cite so many fragments of verses (most famously "avadim hayinu")? The Sfat Emet claims that it is not considered to be breaking a verse if the verse contains the words "le'emor" or "v'amarta" – "and you should say" or "so saying." Since the verse describes something tat one should say, one only has to say that part, and does not have to recite as well the command to say it. The Tzitz Eliezer gives a more technical answer, claiming that when the hagadah cites such verses, it is sure to alter a word or two so as to avoid this problem."
Thank you to @Avraham's comment, I realized I need to mention that there are exceptions for this rule, and one must take each case individually. Here are some of the exceptions.