(Revised) In principle, they obtained this capability in 1951.
In 1949 the Soviets introduced Tu-4 (an unlicensed copy of B-29, by "reverse engineering") with range 5400 km and 847 of them were built in 1949-1952. It was the first Soviet aircraft capable to drop nuclear weapon. With this range it could reach major US cities without return.
The Soviets were prompt in demonstration of this capability. Tu-4 was shown to the world in an aviation demonstration in August 1947 (4 of them appeared).
The next model was Tu-16 (1500 built starting 1954) with range 7200 km,
already sufficient to hit NY from Murmansk region and come back.
However, Yuly Khariton in his memoirs mentions that the first air dropped
bomb was tested in October 1951, from Tu-4. (The earlier bombs were not designed for air drop).
Another reference addressing exactly this question is
CIA. Special National Intelligence Estimate. Soviet Capabilities for Attack on the US through 1957. SNIE 11–2-54, published February 24, 1954. P. 5–6. HSTL, PSF.
EDIT2. A good source is the book of David Holloway, Stalin and the bomb. He cites the following dialog about M-4 bomber by Miasishchev (1954):
"When Khrushchev expressed his dissatisfaction with M-4 having no sufficient range, Miasishchev replied that the plane can bomb in the US and then land in Mexico. Khrushchev: Do you consider Mexico our mother-in-law!? Whom you can visit at any time? Mexicans will never return our plane".
Very typical of Khrushchev:-)
BTW Holloway estimated that SU had only 50 nuclear bombs by 1954.