Inspired by works of Castéras (Pacific J. Math., 2015), Li-Zhu (Calc. Var., 2019) and Sun-Zhu (Calc. Var., 2020), we propose a heat flow for the mean field equation on a connected finite graph $G=(V,E)$. Namely $$ \left\{\begin{array}{lll} \partial_tφ(u)=Δu-Q+ρ\frac{e^u}{\int_Ve^udμ}\\[1.5ex] u(\cdot,0)=u_0, \end{array}\right. $$ where $Δ$ is the standard graph Laplacian, $ρ$ is a real number, $Q:V\rightarrow\mathbb{R}$ is a function satisfying $\int_VQdμ=ρ$, and $φ:\mathbb{R}\rightarrow\mathbb{R}$ is one of certain smooth functions including $φ(s)=e^s$. We prove that for any initial data $u_0$ and any $ρ\in\mathbb{R}$, there exists a unique solution $u:V\times[0,+\infty)\rightarrow\mathbb{R}$ of the above heat flow; moreover, $u(x,t)$ converges to some function $u_\infty:V\rightarrow\mathbb{R}$ uniformly in $x\in V$ as $t\rightarrow+\infty$, and $u_\infty$ is a solution of the mean field equation $$Δu_\infty-Q+ρ\frac{e^{u_\infty}}{\int_Ve^{u_\infty}dμ}=0.$$ Though $G$ is a finite graph, this result is still unexpected, even in the special case $Q\equiv 0$. Our approach reads as follows: the short time existence of the heat flow follows from the ODE theory; various integral estimates give its long time existence; moreover we establish a Lojasiewicz-Simon type inequality and use it to conclude the convergence of the heat flow.