Town saw off competition from Championship and League One sides to land their man, who has netted 23 times for the Knitters this season.
The 20-year-old has joined on loan until the end of the campaign, when he will agree a permanent deal worth £30,000.
Hatters have already come up against Gray twice this year in the FA Trophy when he was denied a goal by two stunning performances from keeper Mark Tyler.
Manager Gary Brabin said: “Andre is just what we’re looking for. He showed us in the two games against us what he’s all about: he’s quick, strong and direct.
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“He gives us exactly what we are lacking right now; a real goal threat, and has proved that, despite Hinckley position in their league, he can still score goals.
“He’s just 20 and while he’s ready now he’s got an enormous amount of potential to be a very good player in the future.
“He was released by Shrewsbury and has bounced back to be an important part of the team at Hinckley. He’s hungry to do well for us and test himself at a higher level.
“Along with our scouts, Alan Neilson, Paul Carden and myself have watched him several times since we played Hinckley in the Trophy and on numerous occasions we’ve all been incredibly impressed with him.
“It’s not just his goal record and speed that stands out but also his work-rate: he never gives the defences a moment’s rest.”
Gray will wear squad number 27 and will be available to play at home to Grimsby Town on Saturday.
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