Henrik Zetterberg added a goal and three assists, and Dan Cleary got the other goal for Detroit. Nicklas Lidstrom had three assists, and Dominik Hasek stopped 24 shots.
Lee Stempniak, Ryan Johnson, and Keith Tkachuk scored for St. Louis. Manny Legace made 24 saves.
Draper knocked Lidstrom's pass, which was tipped by Stempniak, out of the air and into the net with 6:29 left to finish off a 2-on-1 break. Cleary was in the penalty box for goaltender interference.
Cleary got added his 20th goal, into an empty net, with 28 seconds left.
Tkachuk tied it 3-3 with his 16th goal 9:56 into the third period.
Holmstrom's second goal of the game had given Detroit the lead just 2:14 earlier. He scored his 18th goal on a breakaway.
Johnson tied the game 2-2 with 5:19 left in the second period. He sent a pass from Dallas Drake under Hasek. Drake had stolen the puck from Pavel Datsyuk just inside the St. Louis blue-line.
Holmstrom gave the Red Wings a 2-1 lead with a power-play goal 8:29 into the middle period. He redirected in a shot from the point by Lidstrom. Johnson was in the penalty box for goalie interference. He knocked Hasek to the ice after the veteran goalie stopped Johnson's backhand as he drove to the net.
Each team scored on its first shot.
Zetterberg gave Detroit a 1-0 lead 33 seconds into the game with his 21st goal. He cut in off left wing, and put a backhand shot under the crossbar. Stempniak tied it 33 seconds later when he beat Hasek from the right circle with his 16th goal.
Notes: Detroit D Mathieu Schneider returned after missing 10 games with a strained right knee ligament. ... St. Louis RW Bill Guerin played his 1,000th game.