You can cite Planck Collaboration et al. (2016).
This paper describes the latest result from the Planck satellite, which measures the anisotropies of the CMB and infers, among other things, the densities of the various components of the Universe. The density of baryons is measured to be $\Omega_\mathrm{b} = 0.0484$ and the helium fraction $Y = 0.2453$. Ignoring the small correction from metals$^\dagger$, the hydrogen fraction is then $X=1-Y=0.7547$, and the mean atomic mass is
$$
\bar{m} = (X + 4Y) \, m_\mathrm{H} = 1.718\, m_\mathrm{H}.
$$
The Hubble parameter is $H_0 = 67.81\,\mathrm{km}\,\mathrm{s}^{-1}\,\mathrm{Mpc}^{-1}$, yielding a critical density of $\rho_\mathrm{c} \equiv 3H^2/8\pi G = 8.638\times10^{-30}\,\mathrm{g}\,\mathrm{cm}^{-3}$.
These numbers together give you the number density of atoms of
$$
n = \Omega_\mathrm{b}\,\frac{\rho_\mathrm{c}}{\bar{m}} = 1.44\times10^{-7}\,\mathrm{cm}^{-3}.
$$
Using the rest of the cosmological parameters and integrating the Friedmann equation gives the distance to the particle horizon, $R = 14.45\,\mathrm{Gpc}$, corresponding to a volume of $V = 4\pi R^3/3 = 3.71\times10^{86}\,\mathrm{cm}^{3}$.
Thus, the total number of atoms in the observable Universe is
$$
N = n\,V \simeq 5.3\times10^{79}.
$$
$^\dagger$"Metals" means anything heavier than helium. The Planck observations do not deal with metals, which increase the mean atomic mass a little, and hence decrease the total number of atoms. However, firstly the metallicity $Z=1-X-Y$ is small ($Z\simeq0.02$) in galaxies, secondly most of the metals are in galaxies, and thirdly most of the atomic mass is outside the galaxies in the intergalactic medium (see Calura & Matteucci 2004), so the average cosmic metallicity is more like $Z\sim0.01$, increasing $\bar{m}$ to perhaps $1.8\,m_\mathrm{H}$, which in turn decreases $N$ to $5.0\times10^{79}$. These numbers are a bit uncertain because measurements of the metallicity in the IGM are difficult due to the high ionization state of the atoms.
