Regin restored the Senators' 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 Ottawa up 2-0.
Mike Fisher scored a power-play goal 16:45 into the third to seal the win and extend the Senators' streak, which included a 5-1 win in Boston on Monday as Alfredsson had a hat trick.
"I think the last four games we've played well and different lines and different guys have contributed to the team," Regin said. "Last time it was Alfie's line that scored all the goals and tonight we had two big ones so it's good for the team that different guys can score goals."
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.
"It was just one of those nights where we weren't competing and we weren't skating," Blackhawks defenseman Duncan Keith said. "I don't know why but we know that Thursday's game [in Calgary] is going to be tough so we better be ready for that one."
Huet stopped 14 shots.
"If you take away the goals and watch the game, I think they had more puck possession and probably the better chances as well," Alfredsson said. "But our D did a great job in front of the goalie, he made the first saves, we cleared the puck really well and scored some timely goals."
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.
"It's one of those things where he has the whole net to shoot at and you just try to get your stick in the middle of the net," Elliott said. "There's not really too much skill, just hope and determination."
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.
"It's unfortunate but it's not the story of the game," Toews said. "It's not like we outplayed them or played our best game and didn't get a break. That's going to happen sometimes. We've had games this year where we outplayed the other team and deserved to win and we didn't but that wasn't it tonight. We definitely know that there's a lot better in us."
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.
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 Alexei 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.
Copyright by STATS LLC and The Associated Press
Arena: Canadian Tire Centre Location: Ottawa, Ontario
Referees: Stephane Auger, Kevin Pollock Linesmen: Greg Devorski, Scott Driscoll
Attendance: 17,556 (94.9% full)
Team Stat Comparison
Power Play Goals
Power Play Opportunities
Shots on Goal
Zack Smith (2)
Assist: Chris Neil
Ryan Shannon (5)
Assists: Peter Regin, Alex Kovalev
Marian Hossa (13)
(Shorthanded) Assists: Duncan Keith, Brent Seabrook
Peter Regin (7)
Assist: Chris Neil
Mike Fisher (17)
(Power Play) Assists: Nick Foligno, Erik Karlsson