Do all of these parts work together?

They will all work together, but I would use a better PSU.

PCPartPicker part list / Price breakdown by merchant

CPU: Intel Core i5-4590 3.3GHz Quad-Core Processor ($188.99 @ SuperBiiz)
Motherboard: Gigabyte GA-H97M-HD3 Micro ATX LGA1150 Motherboard ($78.99 @ Amazon)
Memory: Kingston HyperX Fury Black 8GB (2 x 4GB) DDR3-1600 Memory ($37.99 @ Amazon)
Storage: Western Digital Caviar Blue 1TB 3.5" 7200RPM Internal Hard Drive ($49.89 @ OutletPC)
Video Card: Gigabyte Radeon R9 380 4GB SOC Video Card ($198.98 @ Newegg)
Case: Corsair SPEC-02 ATX Mid Tower Case ($47.99 @ Amazon)
Power Supply: EVGA SuperNOVA G2 650W 80+ Gold Certified Fully-Modular ATX Power Supply ($69.99 @ Newegg)
Optical Drive: Asus DRW-24B1ST/BLK/B/AS DVD/CD Writer ($15.99 @ Newegg)
Operating System: Microsoft Windows 10 Pro OEM (64-bit) ($124.74 @ B&H)
Total: $813.55
Prices include shipping, taxes, and discounts when available
Generated by PCPartPicker 2016-02-09 22:42 EST-0500

http://www.tomshardware.com/forum/id-2547993/psu-tier-list.html
 
Solution
No budget for an SSD as well? Won't make any difference to your gaming, but for general responsiveness of the computer for everyday tasks, it makes a massive difference. You'd need to add at least $50 to your budget though for an entry level model, or better still $68 for a 120GB 850 EVO: http://pcpartpicker.com/part/samsung-internal-hard-drive-mz75e120bam
Or $85 for the 250GB model: http://pcpartpicker.com/part/samsung-internal-hard-drive-mz75e250bam

It's not essential, but would make a big difference if you can squeeze it into your budget.