Anderson, a backup for the Panthers from 2006-07 through 2008-09, improved to 9-0-1 against his former team. He shut out Florida 4-0 Monday night at Ottawa.
With two victories against the Panthers this season, the Senators have taken 11 of the past 12 meetings dating to the 2009-10 season.
Ottawa is off to a 3-0 start for the first time since 2007-08 when it opened with five consecutive victories.
Tomas Fleischmann scored for the injury-riddled Panthers, who have lost three in a row since opening the season with a 5-1 victory against the Carolina Hurricanes on Saturday night. Florida has scored two goals in its three losses.
Already missing regulars Kris Versteeg, Marcel Goc, Sean Bergenheim and Erik Gudbranson, the Panthers also found themselves without center Stephen Weiss after he was scratched with a lower-body injury.
Jose Theodore finished with 34 saves, including 14 in the first 11 minutes before Phillips beat him with a slap shot from the point toward the end of a second consecutive power play.
Theodore stopped Guillaume Latendresse's penalty shot with 9:11 left in the third period. Latendresse was awarded the penalty shot after he was hooked from behind by Brian Campbell while on a breakaway.
Latendresse tried to slip a quick backhand through the five-hole on a deke, but Theodore made a stick save.
Karlsson broke the 1-1 tie at 8:28 of the second period after a wild sequence in front of the Florida net. After a loose puck bounced in front of Theodore, three players wound up on the ice before the Panthers gained possession.
Karlsson intercepted Shawn Matthias' cross-ice pass just outside the blue line, and after waiting for a teammate to clear the zone, skated in down the left side before beating Theodore with a short-distance wrist shot to the far side.
It was the Norris Trophy winner's second goal of the season.
Fleischmann opened the scoring at 8:24 with a power-play goal. After Tomas Kopecky controlled the faceoff, Fleischmann passed the puck back to Campbell at the point. Campbell flipped it back to Fleischmann, who skated in then beat Anderson with a wrist shot through the five-hole from the top of the circle.
The goal snapped Anderson's shutout streak at 122:24. He hadn't given up a goal since the 6:00 mark of the first period of the opener when Winnipeg Jets defenseman Dustin Byfuglien scored on a power play.
Anderson wasn't tested very often in the third period Thursday, but he came up big when he gloved Peter Mueller's shot from the wing with 6:07 left.
Seconds earlier, Jakob Silfverberg almost increased Ottawa's lead but his wrist shot bounced off the right post.