He was very tired and fell asleep with the TV o[ ]

这是with的用法

“with+宾语+形容词或短语”，如：

1，With so much water vapour present in the room, some iron-made utensils have become rusty easily.

2， Every night, Helen sleeps with all the windows open.

on
表示一种状态。For example,He fell asleep with the windows open.这句话中的open用法一样的。

他非常累，在看电视是睡着了。