The statement is false, here is a counterexample. First note that for a Lie Group $G$ and its closed subgroup $H$, we have a fibration $G/H\to BG\to BH$. $BG$ and $BH$ are not finite, but they are almost just as good, if you really want finite CW complex, you can restrict the fibration over a finite skelton of $BH$. Now, let's take $G=S^3$, $H=S^1$. Then $G/H$ is homeomorphic to $S^2$. If we restrict this fibration to the $2$-skelton of $BH$, one gets the example mentioned in the comment by @Gustavo Granja, but let's look at the whole thing. We have $BG=CP^{\infty }$, $BH=HP^{\infty }$ so, the $E^2$ term of the spectral sequence is $$P[z]\otimes \Lambda (y)$$ with $y$ in degree 2, $z$ in degree 4, whereas the cohomology of the total space is $P[x]$ with $x$ in degre 2.

The statement is false, here is a counterexample. First note that for a Lie Group $G$ and its closed subgroup $H$, we have a fibration $G/H\to BH\to BG$. $BG$ and $BH$ are not finite, but they are almost just as good, if you really want finite CW complex, you can restrict the fibration over a finite skelton of $BH$. Now, let's take $G=S^3$, $H=S^1$. Then $G/H$ is homeomorphic to $S^2$. If we restrict this fibration to the $2$-skelton of $BH$, one gets the example mentioned in the comment by @Gustavo Granja, but let's look at the whole thing. We have $BG=CP^{\infty }$, $BH=HP^{\infty }$ so, the $E^2$ term of the spectral sequence is $$P[z]\otimes \Lambda (y)$$ with $y$ in degree 2, $z$ in degree 4, whereas the cohomology of the total space is $P[x]$ with $x$ in degre 2.