This is not quite the form you requested. But if your goal was to control chi-squared left tails, I have found these lower bounds helpful.

**For $X_1 \sim \chi^2_1$:**

Let $Z \sim N(0,1)$, so that $Z/\sqrt{2} \sim N(0, 1/2)$.
For $x>0$, we have $\text{erf}(x) = P\left(|Z/\sqrt{2}| \leq x\right) = P(X_1 \leq 2x)$.

Apply [this bound](http://math.stackexchange.com/questions/6828/proving-sqrt1-x2-ge-operatornameerf-sqrt-log-x) or [this refinement](http://math.stackexchange.com/questions/6908/proving-1-exp-4x2-pi-ge-texterfx2): $\text{erf}(x) \leq \sqrt{1-\exp(-4x^2/\pi)}$.

Then $P(X_1 \geq y) \geq 1- \sqrt{1-\exp(-2y/\pi)}$ for $y>0$.

**For $X_2 \sim \chi^2_2$:**

Directly from the [chi-squared CDF](https://en.wikipedia.org/wiki/Chi-squared_distribution#Cumulative_distribution_function) for $n=2$, we see that $P(X_2 \geq x) = e^{-x/2}$.

Also, since the chi-squared distribution shifts right as the df increases, $P(X_n \geq x) \geq e^{-x/2}$ for $n\geq 2$.

**For other $n$:**

See [this answer](http://mathoverflow.net/questions/122868/what-is-the-order-of-the-lower-tail-of-a-chi-squared-distribution) to another question.