Robby Fabbri scored for the Red Wings (9-28-3), who have lost 18 of their past 20 games (2-16-2). Bernier made 37 saves after missing the previous two games because of a lower-body injury.
"I wish I had the answer (for the losses), but at least we're competing out there and fighting for it," Red Wings center Frans Nielsen said. "We're giving ourselves a chance. We've got to keep this mentality and we'll be getting some wins, too."
Palat gave Tampa Bay a 1-0 lead at 4:10 of the second period on a redirection of Kevin Shattenkirk's shot.
"I won the face-off and [Shattenkirk] just shot it," Palat said. "I went to the net and was just able to get a stick on it and was able to deflect."
Fabbri tied it 1-1 at 17:54, redirecting Filip Hronek's shot that was going wide short side on McElhinney.
"I thought our guys came out and played hard. We battled, but we didn't have enough," Red Wings coach Jeff Blashill said. "The ice got tilted the wrong way. We lost momentum on our power play in the first period after a real good start."
Luke Glendening had a chance to give Detroit the lead early in the third period on a shorthanded breakaway but was denied by McElhinney.
"I think the compete level was real good. The sacrifice level was real good," Blashill said. "We're getting a little bit better in some areas. What we can't do is let these losses wear us down. It's a mental test for sure, but we've got to find a way to not worry about the last game and just worry about the game ahead of us."