The visitors took the lead in the 16th minute when Andersen raided through the middle and powered an effort past Joel Drommel into the net.
Andersen doubled his and the team's tally for the evening as he was afforded space and time on the edge of the box to drill his effort into the top corner.
However, Andersen completed his hat-trick in stoppage time in the first half by linking up with Van der Velden before slotting his effort past Drommel.
In the second half, Van der Velden almost notched a fourth goal for the visitors but he put his strike from inside the box wide of the post.
Twente pushed to get back into the game with Renato Tapia coming closest to firing them back into the match, but Willem proved to be resolute at the back to hold out for the three points.
The result lifts Willem above Twente into 15th place in the table, while the home side drop down to 16th.