Cooper, 45, compiled a 133-62-26 record over three seasons while guiding the Lightning's American Hockey League club.
He takes over from Guy Boucher, who was fired Sunday.
Assistant coaches Martin Raymond and Dan Lacroix handled the coaching duties Sunday in the Bolts' 3-2 road loss to the Winnipeg Jets.
"[Cooper] has had success at every level he has coached and is extremely familiar with our organization, as well as our players," General Manager Steve Yzerman said. "He has a tremendous record at all levels and we feel he is ready to make the move to the NHL."
Cooper takes over as the Lightning are in danger of missing the post-season after compiling a 13-18-1 mark. They are six points out of a playoff spot.