"He's been huge for us," Lightning coach Jon Cooper said. "We haven't always been great during these six games, but Andrei has kept us alive when we have a bad stretch."
Justin Abdelkader and Mike Green scored for the Red Wings (4-2-0), who got 23 saves from Jimmy Howard.
Kucherov scored his second of the night at 1:47 of the third period to put the Lightning ahead 3-2. With the puck loose at the edge of the crease, he lunged to tip it between Howard's legs.
"I went to cover the puck, and he poked it past me," Howard said. "He always seems to get timely goals against us. We've got to change that."
The Lightning were ahead 2-0 after the first period but couldn't hold on to the lead.
Johnson gave Tampa Bay a 1-0 lead, skating past Detroit defenseman Trevor Daley and breaking in to beat Howard at 9:15 while the Lightning were killing a Chris Kunitz cross-checking penalty against Abdelkader.
Kucherov extended his scoring streak and made it 2-0 by beating Howard for a power-play goal at 10:56.
"I thought we were the better team 5-on-5, but they got those two on special teams and we didn't get any," Red Wings coach Jeff Blashill said.
Detroit was 0-for-6 on the power play.
"We should have had a couple power-play goals," Red Wings captain Henrik Zetterberg said. "We had the chances but didn't finish."
Abdelkader scored on a penalty shot at 2:30 of the second period to get Detroit within 2-1 after being hooked by Tampa Bay defenseman Victor Hedman on a breakaway.