Let $X$ be an infinite-activity isotropic unimodal Lévy process in $\Bbb{R}^d$, with no Gaussian part. Let $\Gamma$ be a cone with a vertex at the origin, and let $\tau_\Gamma$ denote the first exit time of $X$ from $\Gamma$. Let $\nu(r) \coloneq \nu (x)$; if $|x| = r$, denote the radial Lévy density, and assume the following: (i) there exists $\beta \in (0,2)$ and $M > 1$ such that $$ \frac{\nu(r_1)}{\nu(r_2)} \leq M \left( \frac{r_1}{r_2} \right)^{-d-\beta}; $$ (ii) $\nu(r)$ is regularly varying at infinity with index $-d-\alpha$, for some $\alpha \in (0,2)$; (iii) $e^{-t_0 \psi}$ is integrable for some $t_0 > 0$, where $\psi$ is the characteristic exponent of $X$. \par The main result of the paper is the following Yaglom-type limit theorem. There exists a measure $\mu^\alpha$ on $\Gamma$ such that for any $x \in \Gamma$ and $A \subset \Gamma$, $$ \lim_{t \to \infty} \Bbb{P}_x ( a(t) X_t \in A \mid \tau_\Gamma > t) = \mu^\alpha(A), $$ where $a(t)$ is an explicitly given norming function. The limit measure $\mu^\alpha$ only depends on the cone $\Gamma$, and on the index of regular variation, showing the {\it universality} of the Yaglom limit. \par The theorem above extends the results in [K. Bogdan, Z. Palmowski and L. Wang, Electron. J. Probab. {\bf 23} (2018), Paper No. 11; MR3771748], where isotropic $\alpha$-stable processes were treated.