We describe the critical window for percolation in the universality class of sparse growing random graphs. In our models, vertices arrive sequentially and connect independently to each earlier vertex $v$ with probability proportional to a nonpositive power of the arrival time of $v$, continuing until the graph has $n$ vertices. This class includes uniformly grown random graphs and inhomogeneous random graphs of preferential-attachment type. Whenever the critical percolation threshold is positive, we show that the critical window has width of order $(\log n)^{-2}$ and a secondary phase transition at its finite upper boundary. Inside this window the largest component has size of order $\sqrt{n}/\log n$, and the susceptibility remains finite and independent of the position in the window. The proofs couple component explorations to branching random walks killed outside an interval of length $\log n$, allowing sharp control of the barely subcritical and critical regimes.