Regin restored Ottawa's two-goal lead late in the second when he beat Chicago's Cristobal Huet on a wraparound after Zack Smith and Ryan Shannon scored in the first to put the Senators up 2-0.
Mike Fisher scored a power-play goal 16:45 into the third to seal the win and extend Ottawa's streak.
Marian Hossa scored his third short-handed goal of the season earlier in the second for the Blackhawks, who ended a three-game winning streak.
Chicago, which held a 30-18 advantage in shots, entered tied with San Jose for No. 1 overall in the NHL with 72 points after beginning a season-high eight-game trip with wins in Columbus and Detroit over the weekend.
Huet stopped 14 shots.
Hossa, who spent his first seven seasons with Ottawa, drove in on a breakaway and sent a slap shot past Elliott for his 13th goal 6:05 into the second while Blackhawks captain Jonathan Toews was in the penalty box.
Chicago, which entered tied with the Sharks for the league lead in short-handed goals, became the first team to reach eight this season.
Regin, who assisted on Shannon's goal 12:31 in, restored Ottawa's two-goal margin when he drove the right side and went around the back of the net to put a backhand into an open left side with 1:55 left in the second.
Elliott came up with a big stop on Patrick Kane's shot off a rebound with just over 4 minutes remaining in the third. Elliott survived a close call earlier in the third when he recovered to make a sprawling stick save on Hossa just over 6 minutes into the third after he misplayed a puck behind his net.
Hossa had a wide open net staring at him when Elliott inadvertently directed a dump-in toward his vacated crease. The Senators' goalie lunged and stuck out his stick to deny Hossa's shot.
Smith scored his second goal in three games at 6:51 of the first.
Shannon made it 2-0 at 12:31 when he redirected Regin's centering pass from the left side beyond Huet for his fifth goal.
Notes
Hossa, who was chosen 12th overall by the Senators in the 1997 draft, had 188 goals and 202 assists for 390 points in 467 regular-season games with Ottawa from 1997-98 to 2003-04. … Senators RW Alex Kovalev extended his points streak to a season-high four games with an assist on Shannon's goal. He has a goal and three assists during that span. … Ottawa RW Chris Neil had two assists.