You should have no problem, assuming you're not attempting to play above 1080p.
The minimum requirements for the game are a Core2Duo E4400 - the i3-4160 is more than capable.
Honestly, the i3 is capable of running more powerful cards than the 7870, I wouldn't bother upgrading to a Haswell i5. The only real worthwhile option would be the 4690K, but you'd need a new motherboard to OC it. On a budget, the 4160 is still a great option.
Out of curiosity, why do you have 24GB RAM? That's totally overkill for gaming, and the i3 isn't exactly the best option if you're getting into RAM intensive tasks like editing etc.