Curtis McElhinney made 15 saves for the Lightning (34-14-3), who had won four straight.
"We've still got a few games left before we jump into the real hockey," he said.
Tampa Bay fell two points behind the Florida Panthers for second in the eight-team Central Division, and four points behind the first-place Carolina Hurricanes. Florida and Carolina each won Saturday.
Each of the three teams has clinched a Stanley Cup Playoff berth.
"I like the way we've been playing, and we did everything we could to win that hockey game," Lightning coach Jon Cooper said. "Every point is magnified right now, but I'd happily take that game all over again."
Thomas Greiss made 33 saves for his second straight shutout for the Red Wings (18-26-9), who had lost three straight. Greiss, whose shutouts are the only two the Red Wings have this season, has not allowed a goal in 136:31.
"I'm feeling good, but I'm also getting some breaks right now," he said. "They had a couple 2-on-1s tonight where they couldn't put the puck in an open net. That's what you need."