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We constrain the matter density $\Omega_{\mathrm{m}}$ and the amplitude ofdensity fluctuations $\sigma_8$ within the $\Lambda$CDM cosmological model withshear peak statistics and angular convergence power spectra using mass mapsconstructed from the first three years of data of the Dark Energy Survey (DESY3). We use tomographic shear peak statistics, including cross-peaks: peakcounts calculated on maps created by taking a harmonic space product of theconvergence of two tomographic redshift bins. Our analysis follows aforward-modelling scheme to create a likelihood of these statistics usingN-body simulations, using a Gaussian process emulator. We include the followinglensing systematics: multiplicative shear bias, photometric redshiftuncertainty, and galaxy intrinsic alignment. Stringent scale cuts are appliedto avoid biases from unmodelled baryonic physics. We find that the additionalnon-Gaussian information leads to a tightening of the constraints on thestructure growth parameter yielding$S_8~\equiv~\sigma_8\sqrt{\Omega_{\mathrm{m}}/0.3}~=~0.797_{-0.013}^{+0.015}$(68% confidence limits), with a precision of 1.8%, an improvement of ~38%compared to the angular power spectra only case. The results obtained with theangular power spectra and peak counts are found to be in agreement with eachother and no significant difference in $S_8$ is recorded. We find a mildtension of $1.5 \thinspace \sigma$ between our study and the results fromPlanck 2018, with our analysis yielding a lower $S_8$. Furthermore, we observethat the combination of angular power spectra and tomographic peak countsbreaks the degeneracy between galaxy intrinsic alignment $A_{\mathrm{IA}}$ and$S_8$, improving cosmological constraints. We run a suite of tests concludingthat our results are robust and consistent with the results from other studiesusing DES Y3 data.

Dark Energy Survey Year 3 results: Cosmology with peaks using an emulator approach