He threw the staff down in front of Pharaoh, and what landed was not a staff.
וַיַּשְׁלֵךְ אַהֲרֹן אֶת מַטֵּהוּ לִפְנֵי פַרְעֹה וְלִפְנֵי עֲבָדָיו וַיְהִי לְתַנִּין׃
שמות ז:י
"And Aharon cast down his staff before Pharaoh and before his servants, and it became a tanin."
Shemos 7:10
Egypt could do it too. That was the whole point of calling them in.
וַיִּקְרָא גַּם פַּרְעֹה לַחֲכָמִים וְלַמְכַשְּׁפִים וַיַּעֲשׂוּ גַם הֵם חַרְטֻמֵּי מִצְרַיִם בְּלַהֲטֵיהֶם כֵּן׃
שמות ז:יא
"And Pharaoh too called for the wise men and the sorcerers, and the chartumim of Egypt did the same with their spells."
Shemos 7:11
Then the staff swallowed theirs. Not the snake. The staff.
וַיִּבְלַע מַטֵּה אַהֲרֹן אֶת מַטֹּתָם׃
שמות ז:יב
"And the staff of Aharon swallowed their staffs."
Shemos 7:12