This is interesting! I don't know much about Esdeath, as I've only just looked through her profile, but it seems like a good matchup.
So, Esdeath seems to have the AP advantage, though I can't say for 100% certain. Even so, it should be notable.
One thing that should be mentioned is that most of Esdeath's abilities are based around ice, seemingly, which puts them at a large disadvantage. Dante has some pretty good resistance to ice, having resistance to extreme cold and having experience fighting characters with similar abilities.
Dante also has a massive versatility advantage, with a wide variety of varied attacks and abilities that Esdeath does not have any resistance to, such as time slow, BFR, resurrection, healing, duplication (meaning Esdeath would have to fight two of him) and the overpowered goddess that is Royal Guard Attack Reflection.
Esdeath has Time Stop, which could definitely be a problem, however it seems she can only use it for a short time once a day before exhausting herself, and Dante's Mid Regen should be able to regenerate any serious damage she does once she uses that ability. Also, Dante's Time Slow can be spammed instead.
It would appear that Esdeath has the skill advantage, and I'd imagine quite a notable one. They are stated to be in the same intelligence category, and Dante is undeniably incredibly skilled in combat, but I'd still say Esdeath has a clear advantage in that category.
So Esdeath has a notable AP and skill advantage, but Dante seems to quite easily counteract most of Esdeaths abilities while following up with incredibly overpowered hax and abilities for his tier that Esdeath does not have any good resistance against. This includes being able to reflect all of Esdeath's attacks back at them, slowing down time, going Devil Trigger to get a massive AP, Durability and Speed advantage, duplicating himself to force Esdeath to fight two of him at the same time, regenerating from Esdeath's major attacks, and resurrecting in the event that Esdeath manages to kill him.
I'd consider this a high difficulty win for Dante. The AP and Skill advantage would be tricky to bypass, but Esdeath can't do much against Dante's better hax.