As of my last update, IMSLP contains several works by (1903–1978), though the collection is not complete due to copyright restrictions in some countries.
Here is the text you can use for a search or reference regarding on IMSLP (International Music Score Library Project / Petrucci Music Library): Khachaturian on IMSLP khachaturian imslp
For his popular ballets and concertos, check commercial editions (Boosey & Hawkes, Sikorski, or Russian publishers) or library subscriptions. As of my last update, IMSLP contains several