Update 2 after @DavidZ said I had not answered the question and quite correctly deleted my answer.
I rewrote my answer but could not undelete it as it was deleted by a moderator so this is the reason for a second answer.
This is an attempt to illustrate the two interpretations of the change in velocity, $d\vec v$ and $|d \vec v|$, which is possibly the reason for the confusion.
Here is a diagram which shows how under the action of a force $\vec F$ for a short interval of time the initial velocity $\vec v$ changes to a final velocity $\vec v + d \vec v$
$\vec v \cdot \Delta \vec v = v \, |\Delta \vec v| \cos \phi$ where $|\Delta \vec v|$ is the magnitude of the change in velocity $\Delta \vec v$
As the time interval over which the acceleration took place tends to zero this equation becomes
$\vec v \cdot d \vec v = v \, |d \vec v| \cos \phi$
The magnitude of the final velocity $|\vec v + \Delta \vec v| = v +\Delta v = v\cos \Delta \phi+ |\Delta \vec v|\cos(\phi - \Delta \phi)$
As the time interval over which the force acts tends to zero $\Delta \phi \rightarrow 0$
$v\cos \Delta \phi \rightarrow v$ and $|\Delta \vec v|\cos(\phi - \Delta \phi) \rightarrow |d \vec v|\cos\phi $ which is $dv$.
Finally the desired result $\vec v \cdot d \vec v = v \, |d \vec v| \cos \phi = v \, dv$