Let $\mathcal{M}$ denote the class of multiplicative functions with values in the unit disk, and, for $x\geq 1$, $y\geq 1$, let $S(x,y)$ designate the set of $y$-friable positive integers not exceeding $x$. We provide, as $x$ and $y$ tend to infinity in prescribed ranges, upper bounds for exponential sums of the form
$$E_f(x,y;\vartheta):=\sum_{n\in S(x,y)}f(n)\hbox{\rm e}^{2\pi i n\vartheta}$$ whenever $f\in \mathcal{M}$ and $\vartheta$ is a rational number with denominator not exceeding a fixed power of $\log x$.