You are sooo close, but beware of what the first parameter of pbinom(3, 6, prob) means: it gives you the probability of <=3 successes, so 1 - pbinom(3, 6, prob) gives you the probability of >3 successes (not the >= desired).
Therefore you need to change the formula to either
1 - pbinom(2, 6, prob) or
sum(dbinom(3:6, 6, prob)) which you may find more intuitive.
Or if you really did want a simple simulation solution you could use this:
mean(rbinom(1000000, 6, prob) >= 3) (run it multiple times to get slightly different estimates)