Since you have access to a uniform random number generator, generating a random number distributed with other distribution whose CDF you know is easy using the inversion method.
So, generate a uniform random number u in [0,1) range, then calculate x by:
x = log(1-u)/(-λ)
x = log(1-uniformRand(0, 1))/(-λ)
where λ is the rate parameter of the exponential distribution. Now, x is a random number with an exponential distribution. Note that log above is ln, the natural logarithm.