The ten Avatars of Vishnu are as under,though there are minor variations in some Puranas to this list.
The list, according to Vishnu Purana.
Matsya,Koorma,Varaha,Nrusimha,Vamana,Rama,Parashurama,Krishna,Balarama and Kalki.
Of these, the first four were in the first yuga,Satya Yuga Sandhi.
Of these four, Matsya is taken as the first Avatar of Vishnu.
The Matsya cautioned Manu of the impending Great Flood , asked him to meditate and move away.
By that time Manu had a family and had a son,though adopted, name Yagnya.
This is an Avatar of Vishnu as he is the personification of Yagnya,Sacrifice.
He is Yagnya,head of Yagnya,cause of Yagnya,carrier of Yagnya offering.
‘Yagyo Yagya-Patir-Yajva Yagyango Yagya-Vahanah
Yagyabhrud Yagyakrud Yagyi Yagyabhrug Yagyasadhanah
Yagyanantakrud Yagyaguhyam Annam Annada Eva Cha
‘Vishnu has been equated to Yajna (“sacrifice”) as early as in the Vedas. The commenter on the Vedas – Sayana describes Vishnu as the lord of Yajna or the sacrificer himself.Even the Bhagvad Gita associates Vishnu to Yajna (sacrifice). Performing sacrifices is considered to equivalent to pleasing Vishnu.The Vishnu Sahasranama (“Thousand names of Vishnu”) also relates Yajna as a name of Vishnu.
Yajneshwara (“Lord of Yajna”) is mentioned as an avatar of the Hindu god Vishnu in the Bhagavata Purana. As Yajna, Vishnu is the embodiment of the Hindu sacrifice ritual or Yajna.He was also the Indra (king of the gods) of the Svayambhuva Manvantara, era of Svayambhuva Manu.
Yajna is the son of Prajapati Ruci and Akuti, the daughter of Svayambhuva Manu – the first Manu (progenitor of mankind).During the period of Svayambhuva Manu (Svayambhuva Manvantara), there was no qualified Indra, the post of the king of Svarga (Heaven) and king of gods. So, Vishnu incarnated as Yajna and held the post of Indra.
Vishnu Purana tells that Yajna had a twin sister named Dakshina (“donation”). Later, Yajna married Dakshina and had twelve sons. These twelve Devas (gods) are collectively called the Yāmas ( https://en.m.wikipedia.org/wiki/Yajna_(avatar)
Despite of the fact that Âkûti had brothers, she was handed over to the great sage Ruci on the condition that the king would get the resultant son. This was settled with the support of religious rites and the consent of his wife.
Ruci, the most powerful great sage who was entrusted with the procreation, in his supreme concentration begot a pair of children by her who were of the greatest spiritual and brahminical strength.
The very powerful son the daughter bore was taken to the home of the very happy Svâyambhuva Manu, while Dakshinâ was kept by Ruci.’
- Yagna and Dakshina got married subsequently and had 12 sons known as Tushita demigods in Swayambu manu’s time (Tosha, Pratosha, Santosha, Bhadra, Sânti, Idaspati, Idhma, Kavi, Vibhu, Svahna, Sudeva and Rocana) Yagna the Indra then. And Marichi head of sages
It is yet another pointer to Sanatana Dharma being present in the south much earlier to Vedic period.
The philosophical explanation of Yagnya follows in another article.
I have noticed that the name Yagnyanarayanan is present in South India , among Brahmins.
As this Avatar preceded Matsya, which took place after Manu had a family,Yagneswara is the first Avatar of Vishnu.