Here's Eisenbud's general version of Noether's normalization:
Let $A$ be a finitely generated algebra over a field $k$ and $I_1 \subseteq \dots \subseteq I_n$ a sequence of proper ideals of $A$. Then there exists algebraically independent elements $\alpha_1,\dots,\alpha_n$ of $A$ such that $A$ is integral over $k[\alpha_1,\dots,\alpha_n]$ and $k[\alpha_1,\dots,\alpha_n]\cap I_i = (\alpha_1,\dots, \alpha_{p_i})$ for $p_i \geq 0$.
Balwant Singh in his book Basic Commutative Algebra uses the following stategy: he first proves the theorem for $A = k[T_1,\dots,T_n]$, a polynomial algebra over $k$; then he deduces the general case from this one. I attach a screenshot of his deduction since it contains a commutative diagram which I don't know how to type here munally:
(note that the map $k[X'_{p_0+1},\dots,X'_p] \to A$ is first induced by exactness as a $k$-linear map; it can then shows the map is in fact a homomorphism of $k$-algebras). I managed to show that $X_{p_0 + 1},\dots, X_p$ are algebraically independent over $k$ and that $A$ is integral over $k[X_{p_0+1},\dots, X_p]$, but failed to prove that they satisfy $k[X_{p_0+1},\dots, X_p]\cap \mathfrak{a}_i = (x_{p_0+1},\dots,x_{m_i})$. It can be shown that $k[X_{p_0+1},\dots, X_p]\cap \mathfrak{a}_i = \phi(k[X_{p_0+1},\dots, X_p])\cap \phi(\mathfrak {a}'_i)$, but the identity $$\phi(k[X_{p_0+1},\dots, X_p])\cap \phi(\mathfrak{a}'_i) = \phi(k[X'_{p_0+1},\dots,X'_p]\cap \mathfrak{a}'_i)$$ doesn't have to hold since $\phi$ is not necessarily injective. Worse yet, from the other end I can only immediately deduce that $$k[X'_{p_0+1},\dots,X'_p]\cap\mathfrak{a}'_i \subseteq (X'_1,\dots, X'_p).$$
Supposedly, exactness should be used somehow.