Arrhenius equation in exothermic reactions
The Arrhenius equation is absolutely applicable to exothermic reactions and it will give results in accordance to Le Chatelier's principle.
Applying the Arrhenius equation for the forward reaction yields:
$$k_\mathrm{f} = A \exp\left(\frac{-E_\mathrm{f}}{RT}\right),$$
applying the Arrhenius equation to backward reaction yields 
$$k_\mathrm{b} = A \exp\left(\frac{-E_\mathrm{b}}{RT}\right).$$

We know that $E_\mathrm{f} -E_\mathrm{b} = \Delta H$. Dividing $k_\mathrm{f}$ by $k_\mathrm{b}$ we obtain: 
 $$\frac{k_\mathrm{f}}{k_b} = A \exp\left(\frac{-\Delta H}{RT}\right).$$
For an exothermic reaction $\Delta H &lt; 0$. So $-\Delta H/(RT)$ is a positive number. As $T$ increases, the exponent decreases and thus $k_\mathrm{f}/k_\mathrm{b}$ decreases, which is nothing but the decrement of the equilibrium constant.
Thus the result is in perfect accordance with Le Chatelier's principle. 

Also, you can apply the van't Hoff's equation which states: 
$$\ln\left(\frac{k_2}{k_1}\right) = \frac{\Delta H}{R}\cdot\left(\frac 1{T_1} - \frac 1{T_2}\right),$$
which also gives the decrease of rate of reaction with increase in temperature as $\Delta H &lt; 0$.