Asmodeus overthrew He Who Was during the Dawn War, using a fragment of the Shard of Evil, retrieved from the Abyss.
So:
The Abyss is created.
Asmodeus ascends.
The Gods defeat the Primordials, ending the Dawn War.
What's not clear to me is when the Abyss was created relative to the beginning of the Dawn War.
This is speculation, as everyone pointed out there is no a set in stone canonical timeline, but by reading the sources:
The Abyss is created after the destruction of the Living Gate (as Tharizdun got the Shard of Pure Evil from the gate, according to the novels). The Living Gate was destroyed before the Dawn War, or in the earliest battles, as the primordials yet were not an unified force and there were no demons in their ranks yet.
Is also of note that many sources say that Tharizdun was in the gods' side during the Dawn War's early phase. Perhaps he even fought against Tabrach-Ti (the first primordial to be slain in the war) meaning Tharizdun was there way after the creation of the Abyss and the destruction of the Lattice of Heaven. Then he is discovered, sealed, and Asmodeus is tasked with safeguarding his prison.
Asmodeus ascending happens way long after this, as sources about the War of Betrayal say Tiamat helped Asmodeus with her abishai devils. Tiamat only can exist after Io's demise, and following the legends of Haramanthur's sacrifice, this happened in the "last years" of the Dawn War.
If you need all the info about the Dawn War, you can read the compiled info pdf I posted in the Piazza a while back ago. However, as everyone said, the info is vague, often contradictory and the timeline is not set in stone except for a few points. So, you can decide what is true and what don't in your version of the War.
Edit:
Link to the aforementioned pdf:
https://drive.google.com/file/d/0B2mLfpEGKv-Sb29tN2tBYVBWZzA/view