The first companion to perform the prayer was Ali (RA).
Tarikh al-Tabari, Vol. 6, Page 81
Narrated Abbad b. Abdallah:

I heard 'Ali saying, "I am the servent of God and the brother of his Messenger, and I am the most righteous one (al-siddiq al-akbar). No one other than I can say this but a liar and an inventor of falsehood. I performed the prayer with the Messenger of God seven years before other men.

Book  : Tarikh at-Tabari
Author  : Muhammad ibn Jarir al-Tabari (224 AH - 310 AH)
English reference  : Vol. 6, Page 81
Arabic reference  : 
Narrated by: Ahmad b. al-Hasan al-Tirmidhi, from 'Ubaydallah b. Musa, from al-'Ala, from al-Minhal b. 'Amr, from 'Abbad b. 'Abdallah.
 
Ali (RA) performed the Prayer immediately after the Prophet was commissioned.
Tarikh al-Tabari, Vol. 6, Page 80
Narrated Jabir ibn Abdullah:

The Prophet was commissioned as a prophet on Monday, and 'Ali performed the prayer on Tuesday.

Book  : Tarikh at-Tabari
Author  : Muhammad ibn Jarir al-Tabari (224 AH - 310 AH)
English reference  : Vol. 6, Page 80
Arabic reference  : 
Narrated by: Zakariyya b. Yahya al-Darir, from 'Abd al-Hamid b.Bahr, from Shakir b. 'Abdallah b. Muhammad b. 'Aqil, from Jabir ibn Abdullah al-Ansari.
 
Ali (RA) was the first Muslim.
Tarikh al-Tabari, Vol. 6, Page 81
Narrated Afif:

During the Jahiliyyah I came to Mecca and stayed with al-Abbas b. Abd al-Muttalib. When the sun came up and rose into the sky, while I was looking at the Ka'bah a young man came up and gazed at the sky. Then he turned to face the Ka'bah and stood facing it. Soon afterwards a youth came and stood on his right, and soon after that a woman came and stood behind them. The young man bowed, and the youth and woman bowed; then the young man stood erect, followed by the youth and the woman, and then the young man prostrated himself, and they did so with him.

 

I said, "'Abbas this is a weighty matter." "It is a weighty matter indeed," he said. "Do you know who this is?" "No", I said. "This is Muhammad b. 'Abdallah b. 'Abd al-Muttalib, my brother's son," he said. "Do you know who this is with him?" "No," I said. "This is 'Ali b. Abi Talib b. 'Abd al-Muttalib, my brother's son," he said. "Do you kow who this woman is who is behind them?" "No," I said. "This is Khadijah bt. Khawaylid, my nephew's wife," he said "My nephew has told me that his Lord, the Lord of the Heaven, has commanded them to do what you see them doing. God's oath, I do not know anyone on the face of the earth but these three who follow this religion."

Book  : Tarikh at-Tabari
Author  : Muhammad ibn Jarir al-Tabari (224 AH - 310 AH)
English reference  : Vol. 6, Page 81
Arabic reference  : 
Narrated by: Muhammad b. 'Ubayd al-Muharibi, from Sa'id b. Khuthaym, from Asad b. 'Abdah al-Balji, from Yahya b. 'Afif, from Afif.