Be the lead wasn't that huge to start with.

Compare the two most obvious milestones; first satellite and first man into space.

Sputnik : 4/10/1957 Explorer 1 : 1/2/1958

Difference of 4 months and 3 days

Yuri Gagarin : 12/4/1961 Alan Shepard : 5/5/1961

Difference of 23 days.

In the timescale of national-scale major projects, a difference of a few months is very very small.

Once the space race started in earnest, the US could outspend the USSR by a considerable margin. A CIA estimate put the Soviet space funding at 2.4 billion dollars in 1963 to 5.5 billion dollars for the US. This gap narrowed in '65, but pulled apart later; the USSR simply could not afford to match NASA's funding.

Sputnik vs Explorer 1 also shows an interesting comparison. Sputnik was heavier, at 83 Kg, but it only contained a radio beacon. While Explorer 1 was only 13.37 KG, it contained on-board scientific instruments made possible by early transistors. Explorer 1 was able to verify the existence of the Van Allen radiation belt. The lead in electronics technology would prove to be vital, and is a gap that the Soviets could never close.

"Luck" was involved to the degree that it always it. The N1, the Soviet counterpart to Saturn V had exceptionally poor luck in addition to being underfunded. Sergei Korolev, the chief designer of N1 suffered a heart attack in 1960, and died in early 1966. The N1 was also a very complicated design that did not have enough funding or time to work out all it's troubles, not helped at all by Korolev's death halfway through development.

After 4 tests failures and not one successful test launch, there was no way the N1 would be ready in time to contest the Saturn V.

Sources

CIA estimate : http://fas.org/irp/cia/product/sovmim66.pdf