The 22-year-old forward has played a key role for the Danish champions over the past two seasons, having been with the club since the age of 14.
Kusk, who boasts seven caps for Denmark, scored 27 goals in 91 appearances for Aalborg, including 12 last term as he helped the club to a first league title in six years.
Having signed for a Twente side who finished third in the Dutch top-flight last season, Kusk is eager to force his way into Alfred Schreuder's starting XI.
"I feel I am now coming to one of the biggest clubs in the Netherlands. I hope to have lots of playing time in the coming season," he told Twente's website.
"I have already spoken several times with the trainer Alfred Schreuder."
Kusk will link up with national team-mate Andreas Bjelland at Twente.
"For me it's important that I can develop as a player in the coming years," he added.
"I saw that Andreas has got better over the last two years, so I'm at the right place."