Victor Hedman and Taylor Raddysh scored for the Lightning (18-6-4), who have won six of seven. Andrei Vasilevskiy made 28 saves.
Drew Doughty and Arvidsson scored for the Kings (12-10-5), who are 3-1-1 in their past five. Quick made 27 saves.
"You always have to look at the game as a whole," Los Angeles coach Todd McLellan said. "We came in the Stanley Cup champs' building pretty short on the back end, playing a solid 64 minutes, and earned a point."
Doughty gave the Kings a 1-0 lead with a power-play goal from the high slot at 9:06 of the first period.
"I thought we had a chance to win this game," Los Angeles defenseman Olli Maatta said. "It's kind of frustrating to lose that way, but I think there are some good things we can take from this."
Hedman tied it 1-1 at 18:01 with a one-timer from outside the right circle off a pass from Steven Stamkos.
"He roped that," Lightning coach Jon Cooper said. "That was a perfect storm. Right in his wheelhouse with no one in front of him."
Arvidsson gave the Kings a 2-1 lead 30 seconds into the second period. Matt Roy shot through traffic and it deflected off Arvidsson's back.
"Without sounding satisfied, because we're not, we came for two points," McLellan said. "It was a well-played, solid game by us."
Raddysh tied it 2-2 at 14:19 when he redirected a shot by Jan Rutta.
"We just kind of found a way to squeeze another one out there," Raddysh said. "You get into overtime there it can go any way with 3-on-3, but the guys have done a pretty good job of sticking with it and you get an opportunity to put it in the net."