Richards scored two goals to lead the Blackhawks to a 4-1 victory over the Kings on Saturday night in a matchup of teams that both won two of the past five Stanley Cup titles.
"Coming in here is a little bittersweet," said Richards, who failed to score against goalie Jonathan Quick in the five-game final series against Los Angeles. "The feeling is, I wish I had done it the last time I was in here. We would have kept the series going."
The 14-year veteran signed a one-year deal with the Blackhawks after the Rangers bought out the final six years of his nine-year, $60 million contract.
"It's obvious that a nine-year deal going to New York is a lot different than a one-year deal coming into a team that won two Cups," Richards said. "I was fortunate enough that I could be picky in the summer and try to find a good fit, and here we are.
"It's funny, because now I'm on the team they beat to get to us."
Daniel Carcillo also scored for Chicago, and Brandon Saad added an empty-net goal with 1:49 remaining. Corey Crawford made 19 saves in helping the Blackhawks end their road trip 5-1.
"Halfway through the trip we were kind of on the bubble, right around a playoff spot, and we talked about trying to gain ground and gain momentum going into the second 20 games," Richards said. "We did a great job of that. We had big tests against good teams and we piled up five out of six, which is huge."
Jordan Nolan scored for Los Angeles, and Quick stopped 21 shots.
Trailing 2-0, the Kings got on the board at 5:53 of the second period when defenseman Jake Muzzin's slap shot from the left point hit the shaft of Nolan's stick and fluttered between Crawford's pads.
It was the first goal in 14 games this season for Nolan, who would have been a healthy scratch if not for Marian Gaborik's upper-body injury.
Chicago regained its two-goal margin at 8:14 of the second on a rush in which four players touched the puck in the Kings end. Richards converted Michal Rozsival's pass at the left of the net for his sixth goal and third in two games.
"It's nice when the team starts rolling and you feel like you're helping," said Richards, who is playing on a line with Kris Versteeg and Patrick Kane. "Every day it feels more like it's my team. It takes a little bit to feel like you're a part of everything on and off the ice."
Carcillo opened the scoring at 1:52 of the first, batting in his own rebound after the Kings won a faceoff in their zone and turned over the puck.
"We didn't have a good first period and we put ourselves behind the eight ball right away," forward Anze Kopitar said. "It doesn't matter who you're playing in this league, catch-up hockey is not easy."
Just 28 seconds after serving Crawford's penalty for tripping Justin Williams at the edge of the crease, Versteeg took the puck away from Quick and Matt Greene behind the net and found Richards at the left of the crease for a one-timer that beat Quick with 2:01 left in the first.
"They're a tough team to come back on because they're a great defending team," Kings coach Darryl Sutter said. "They're the best team in the league through the first two months of the year, and that's clear. If you're going to give them two goals you're probably going to get your butt kicked. That's exactly what happened."
NOTES: Kings D Alec Martinez, who knocked the Blackhawks out of the playoffs with an overtime goal, missed his seventh straight game because of an upper-body injury. ... This was the sixth time this season that Kane and Jonathan Toews both failed to get a point for the Blackhawks, and the first time the team won (1-4-1). ... Chicago's only power-play opportunity came when Robyn Regehr was sent off for hooking with 10:37 left in the third period. ... The Blackhawks entered Saturday averaging the second-fewest penalty minutes in the NHL (7.5) and a league-best penalty-killing percentage of 90.9. They were short-handed twice in each of their previous four games, and three times against the Kings.