This is list is incomplete, entirely subjective, and presented in no particular order. Most of the music here is from games I personally played last year. I'm sure there are other excellent soundtracks from 2013 and early 2014, so let me know if you have any other recommendations!
A lot of the credit for this list goes to the Top Score podcast by Emily Reese, over at Classical MPR. Additional inspiration from Kotaku.
Bioshock Infinite
Elizabeth
http://youtu.be/1zHeZtWHSPU
AD
http://youtu.be/fBhKuTfbPFI
God Only Knows
http://youtu.be/IJFtFBxpr-I
Will the Circle Be Unbroken
https://www.youtube.com/watch?v=0e4Crth_Hb8
Tearaway
The Message
http://youtu.be/zlsNzbuKiLg
The Gopher
http://youtu.be/SDyEt8KaMpg
Don't Starve
Main theme
https://www.youtube.com/watch?v=mGTTGVdWVUs
Assassin's Creed IV
Sea shanties
https://www.youtube.com/watch?v=bx5uegVP-78
Tavern song: "Here's a Health to the Company"
https://www.youtube.com/watch?v=L-lBydebg6w
Flamenco tavern song
https://www.youtube.com/watch?v=e_NwtUWnhLM
All instrumental tavern music tracks
https://www.youtube.com/watch?v=jSgDAwPrOLI
Brothers
The Beginning of a Journey
http://www.youtube.com/watch?v=0U4JuDVvtG0
The Last of Us
Main Theme
http://www.youtube.com/watch?v=cwDkutphzmU
The Quarantine Zone
http://www.youtube.com/watch?v=A3Qvw2ZAxtw
Rayman Legends
Mysterious Swamps
http://youtu.be/SuImkM2-1kc
Mariachi Madness
https://www.youtube.com/watch?v=xRTIr8ek7eM
Babel Tower
http://www.youtube.com/watch?v=aLnClljaVF8
Dive Another Day
http://www.youtube.com/watch?v=17GOn4Qlxtg
The Legend of Zelda: A Link Between Worlds
Lorule Field
https://www.youtube.com/watch?v=OYc_Vosp9_Y
Ni No Kuni Soundtrack
14 World Map ~Another World~
https://www.youtube.com/watch?v=-DKgEZfTyU8
Fire Emblem: Awakening
Don't Speak Her Name
http://www.youtube.com/watch?v=iWO4ff1HXYY
Starbound
On the Beach at Night
https://www.youtube.com/watch?v=Qy57BDR6Y0Q
Tomb Raider
"A Survivor Is Born" Suite [LIVE In Sweden]
http://www.youtube.com/watch?v=-jeRxhu9kYs
SOS Tower [Final Version]
http://www.youtube.com/watch?v=XgzGH1hXkgw