The paper deals with the equation \[ x'(t)=(\tilde A+A(t))x(t)+(\tilde B+B(t))x(t-\tau), \] where \(\tau\) is a positive constant. Here \smallskip \((H_1)\) \(\tilde A,\tilde B\) are constant diagonal matrices \(\tilde A=\text{ diag }\big\{ \tilde a_1,\tilde a_2,\ldots,\tilde a_n\big\},\) \(\tilde B=\text{ diag }\big\{ \tilde b_1,\tilde b_2,\ldots,\tilde b_n\big\},\) such that \((|\tilde a_i|+|\tilde b_i|)\tau<1/e\) and the equation \(\lambda\tilde a_i+\tilde b_i e^{-\lambda\tau}\) has a unique real root \(\tilde\lambda_i\) in the interval \((\tilde a_i-1/\tau,\infty),\) \(i=1,2,\ldots,n.\) These roots do not coincide, i. e. \(\tilde\lambda_i\neq\tilde\lambda_j\) for \(i\neq j,\) \(i,j=1,2,\ldots,n.\) \smallskip \((H_2)\) \(A(\cdot)=(a_{ij}(\cdot))_{1\leq i,j\leq n},\) \(B(\cdot)=(b_{ij}(\cdot))_{1\leq i,j\leq n}\) are continuous matrix functions on \([0,\infty)\) such that \(\alpha(t):=\sup_{s\geq t}|A(s)|\in L^2,\) \(\beta(t):=\sup_{s\geq t}|B(s)|\in L^2,\) and \(a_{ii}(t)-{1\over\tau}\int_{t-\tau}^t a_{ii}(s) ds\in L^1,\) \(b_{ii}(t)-{1\over\tau}\int_{t-\tau}^t b_{ii}(s) ds\in L^1,\) \(i=1,2,\ldots,n.\) \medskip The main result is: Let hypotheses \((H_1)\) and \((H_2)\) be fulfilled. Then there exists a matrix function \(F\), \(F(t)\to0\) as \(t\to\infty,\) such that the following statements are valid. \smallskip \((A)\) For every constant vector \(c\), the function \(x\) defined by \[ x(t)=(I+F(t))\exp\left(\int_0^t\Lambda(s) ds\right)c, \] where \[ \Lambda(t)=\tilde\Lambda+\big(I+(\tilde\Lambda-\tilde A)\tau\big)^{-1} \big(\text{diag}\{A(t)\}+\text{diag}\{B(t)\}e^{-\tilde\Lambda\tau}\big) \] and \(\tilde\Lambda=\text{diag} \{\tilde\lambda_1,\tilde\lambda_2,\ldots,\tilde\lambda_n\},\) is a solution of the considered equation. \smallskip \((B)\) For every solution \(x\) there exist a constant vector \(c\) and a vector function \(f,\) \(f(t)\to0\) as \(t\to\infty\), such that \[ x(t)=(I+F(t))\exp\left(\int_0^t\Lambda(s) ds\right)(c+f(t)) \] and \[ \left|x(t)-(I+F(t))\exp\left(\int_0^t\Lambda(s) ds\right)c\right|e^{t/\tau}\quad\text{ is bounded for}\quad{t\to\infty}. \] \smallskip A theorem of Hartman--Winter establishes the asymptotic form of the fundamental matrix of a system of ordinary differential equations with asymptotically diagonal matrix. An extension of this theorem to a scalar delay differential equation is done by \textit{J. R Haddock} and \textit{R. J. Sacker} [J. Math. Anal. Appl. 76, 328-338 (1980; Zbl 0446.34071)]. The case of a ``quasi-triangular'' delay-system is considered by the first two authors in an earlier paper. The main result of this paper is an extension to the general case.