Q:

Is it fair to say that the Space Race incentivized research in pure mathematics?

I've only heard personal anecdotes about perceived stimuli and economic support to research in pure mathematics in the 1960s, presumably tied to the space race of the time. Is this something we can claim to have been planned by the governments engaged in the race?

A:

Indeed, funding of mathematics, together with other science and engineering fields
experienced a sharp increase between 1957 and 1970. The number of PhD awarded in these fields in USA tripled during this period, and this was the fastest growth for
any period after WWII.
https://www.nsf.gov/statistics/2018/nsf18304/static/report/nsf18304-report.pdf
But the "race to the Moon" was not the primary reason. The primary reason was arms race during the Cold war. The race to the Moon itself, is a byproduct of this arms race. This happened both in Soviet Union and in the US. Soviets demonstrated their success in ICBM development by launching the first artificial satellite in 1957.
This made an enormous impression on American public, and the "space race" began. Much of mathematical research at that time was funded by the US Navy and Air Force, in the same way as it is funded nowadays by NSF. 
In Soviet Union itself, the sharp increase in S and E research was triggered by the race to make a nuclear bomb, and then it was stimulated by
development of new weapons, like ICBM and submarines.
I am a "pure mathematician", and my career began in Soviet Union in the 1970s. Everyone around was strongly encouraged to make some contribution to research directly related to submarine warfare at that time.
Edit. In general, on my opinion, the arms race strongly benefited mathematics, especially in Soviet Union. They made the careers of professor and researcher very attractive by relatively high salaries. They developed a system of mathematical olimpiads to select talented children. The competition for students to enroll in the best math departments was high.
Some top mathematicians were chosen to advise the government, and could influence some decisions. For example, the initiative of the space exploration came from scientists directly involved in defense research. The ideological interference into physics and mathematics was minimal (as compared for example to biology, and related medical and agricultural sciences, not even mentioning humanities which were under complete ideological control). And so on. 

