Luke Hughes had two assists and Jake Allen made 16 saves for the Devils (37-28-6), who have lost three of their past four games. New Jersey is third in the Metropolitan Division, six points ahead of the New York Rangers.
Tkachuk scored on a tip-in from the slot after a pass by Jake Sanderson from the left face-off circle to give the Senators a 1-0 lead at 4:05 of the second period.
"Brady sets the tone by being a competitive guy," Green said. "He loves winning, loves competing, loves his teammates. He's everything you'd want in a young captain."
Batherson then made it 2-0 on a shot from the slot at 4:38.
Nico Hischier scored a power-play goal on a slap shot from the right face-off circle to pull the Devils within 2-1 at 7:13 of the second. Jesper Bratt had an assist on Hischier's goal to set the franchise record for most in a season (61 assists), passing Scott Stevens (60 in 1993-94).
"I think we obviously had a good push in the third because we're desperate to win but I just don't think in the first or second period we're starting desperate enough," Bratt said. "Our puck battles and puck play just wasn't good enough and we had a hard time creating any offense in those first two periods. We weren't physically engaged in the battles to come up with pucks."
Ottawa limited New Jersey to 12 shots through two periods.
"Recognizing urgency and being capable of executing it is a whole other thing," Devils coach Sheldon Keefe said. "It's hard to do it all the time and that's what we got to push through, individually, collectively. It's hard to do it every second night and that's what we're fighting right now. We've lost players (to injury) and we're asking more of different guys and it's hard. We know this period of time will be extremely difficult and, right now, it's getting the better of us."
David Perron pushed it to 3-1 after a one-timer by Claude Giroux from the right circle deflected in off Perron's hip from the slot at 8:44 of the second.