The 15-year-old from Tamil Nadu lost the first game of the opening set but came back strong and used her swift movements and sublime net play to break Lakshmi’s serve in her 6-1, 6-1 triumph on Tuesday.
Ugarte becomes United's fifth major signing of the transfer window after defenders Leny Yoro, Matthijs de Ligt and Noussair Mazraoui and Dutch forward Joshua Zirkzee.
Three of the top female tennis stars - Iga Swiatek, Naomi Osaka and Coco Gauff - have let go of their coaches.
Prime Minister Narendra Modi hosted India's Paris Olympics contingent after the Independence Day celebrations at Red Fort.
The Paralympic Games, making their debut in France, will begin on August 28 with an opening ceremony between the Champs-Elysees and the Place de la Concorde.