Dylan Strome scored, and Subban made 28 saves in the Blackhawks season opener.
The teams will play again here Friday.
"Certainly not the way we wanted to start the year," Chicago coach Jeremy Colliton said. "Gave up too many chances and too many goals. … We did some good things, but our execution was lacking. We couldn't make them pay."
Palat gave Tampa Bay a 1-0 lead with a power-play goal at 10:39 of the first period with a put back off a shot by Hedman.
"The power play clicked a little bit tonight, so that was good to see," Cooper said. "The guys have been getting their legs back with each day. You saw that tonight. We've got better in us. You saw that game pick up as the game went on, and I expect Friday to be a really good matchup."