Paul made it 3-2 at 16:31 when he scored an unassisted goal on a wrist shot from the high slot.
The Capitals appeared make it 4-2 when Carlson knocked in a loose puck at 19:54, but Tampa Bay successfully challenged for offside.
Carlson did make it 4-2 with his second power-play goal of the game at 16:35 of the second period, scoring from the slot off a pass from T.J. Oshie.
"We couldn't get clears, so that first one they spent however long in our zone and eventually buried one," Cooper said of the penalty kill. "And the second one, one of our 'D' took a shot, so he was down and we probably collapsed a little bit too low and clearly Carlson was hot tonight. Everything he was shooting, either he was scoring it or somebody was."
NOTES: Ovechkin is eight points from Dale Hawerchuk for 20th in NHL history. … Kucherov has goals in three straight games and extended his season-best point streak to six games (three goals, eight assists). … Stamkos extended his point streak to eight games (five goals, eight assists). … Ovechkin has 11 goals in his past 15 games. … Capitals forward Nicklas Backstrom had his 743rd NHL assist, passing Denis Potvin for the 45th in NHL history. … Kuznetsov has points in 14 of his past 15 games (seven goals, 11 assists).