In a recent work, Allen, Böttcher, Hàn, Kohayakawa, and Person provided a first general analogue of the blow-up lemma applicable to sparse (pseudo)random graphs thus generalising the classic tool of Komlós, Sárközy, and Szemerédi. Roughly speaking, they showed that with high probability in the random graph $G_{n,p}$ for $p \geq C(\log n/n)^{1/Δ}$, sparse regular pairs behave similarly as complete bipartite graphs with respect to embedding a spanning graph $H$ with $Δ(H) \leq Δ$. However, this is typically only optimal when $Δ\in \{2,3\}$ and $H$ either contains a triangle ($Δ= 2$) or many copies of $K_4$ ($Δ= 3$). We go beyond this barrier for the first time and present a sparse blow-up lemma for cycles $C_{2k-1}, C_{2k}$, for all $k \geq 2$, and densities $p \geq Cn^{-(k-1)/k}$, which is in a way best possible. As an application of our blow-up lemma we fully resolve a question of Nenadov and Škorić regarding resilience of cycle factors in sparse random graphs.