Matt Garza, Josh Johnson and Joe Blanton stayed put for now as the window closed for clubs to make trades without having players pass through waivers.
Four days after the AL West rival Los Angeles Angels acquired former Cy Young Award winner Zack Greinke, the Rangers obtained Dempster for infielder Christian Villanueva and pitcher Kyle Hendricks. Texas also acquired catcher Geovany Soto from the Cubs for right-hander Jake Brigham.
"It's a great opportunity over there," Dempster said. "It's not going to be easy. There are a lot of teams out there that are really good. I think they have as good a chance as anybody."
Dempster, who will replace Roy Oswalt in the rotation, had power to block deals and refused to approve a trade to Atlanta last week.
"I just never said no," he explained. "The last few days, I had to give it a lot more thought to the teams I might possibly be traded to."
Dempster was 5-5 with a 2.25 ERA in 16 starts for the Cubs and can become a free agent after the season.
"He's a veteran. He's been through some wars before," said Rangers general manager Jon Daniels, who team has won two straight AL pennants but no World Series.
The faded Philadelphia Phillies, last in the NL East and their run five straight division titles all but over, sent Victorino to the Los Angeles Dodgers for right-handers Josh Lindblom and Ethan Martin and cash. They also shipped Pence to San Francisco for outfielder Nate Schierholtz, catching prospect Tommy Joseph and right-hander Seth Rosin.
Cincinnati received Broxton from Kansas City, also a cellar dweller.
"When you're in last place, you can try any damn thing," Phillies manager Charlie Manuel said. "You don't have nothing to lose."
The deals capped a busy two-week period that also saw Ichiro Suzuki, Hanley Ramirez, Anibal Sanchez and Francisco Liriano change teams as general managers assessed whether they had a chance to make this year's expanded 10-team playoffs or whether to focus on rebuilding for 2013.
After winning a franchise-record 102 games last year, the Phillies never recovered from Ryan Howard's stumble on the last out of last year's NL division series against St. Louis. Howard, the 2006 NL MVP, tore an Achilles tendon and didn't come off the disabled list until July 6. All-Star second baseman Chase Utley missed the first 76 games because of a chronic problem in both knees.
Pence is batting .271 with 17 homers and 59 RBIs this season. He fills a major need for the Giants, giving them a right-handed hitter with power.
"I don't think anyone really anticipated the season that's gone on," Pence said in Washington after the trade. "It was the perfect storm of injuries and things didn't go right for us, so that's the way the business of the game is and you have to understand that. Everything is understood. The Phillies are going in a different direction. We had a great run at it. Now I'm going a different way."
Victorino, nicknamed the Flyin' Hawaiian, is batting .261 with nine homers, 40 RBIs and 24 steals. He helped the Phillies win five straight NL East titles and the 2008 World Series championship.
"Mahalo to the (at)Phillies and the AMAZING fans in Philly for a great run. A lot of unforgettable memories in this city. I'll miss you guys!" Victorino wrote on Twitter.
While getting rid of stars, the Phillies held on to Cliff Lee and Blanton. After jettisoning Ramirez, Sanchez and Omar Infante as the deadline approached, the Miami Marlins traded pitcher Edward Mujica to St. Louis for infielder and sent former All-Star first baseman Gaby Sanchez and right-hander Kyle Kaminska to Pittsburgh for outfielder Gorkys Hernandez and a 2013 draft pick.
Washington has emerged with young talents Stephen Strasburg and Bryce Harper and leads the NL East as the Nationals try to bring post-season play to the capital for the first time since 1933.
"Our division has undergone a real sort of metamorphosis," Mets general manager Sandy Alderson said.
Houston and the nearly as awful Cubs also shed salaries. Jed Lowrie at $1.15 million is now the highest-paid player on the Astros, who started the season with a payroll of nearly $61 million.
Chicago, in its first season under new baseball head Theo Epstein, also dealt pitcher Paul Maholm and outfielder Reed Johnson to Atlanta on Monday.
NL Central-leading Cincinnati, hopeful of just its second post-season appearance since 1995, got Broxton from Kansas City — the team with the worst record in the AL.
"I'm going somewhere where I'm picking up 20-25 games," Broxton said, noting the standings. "I enjoyed my time here, but this is part of baseball. So go over there and, hopefully we can win a division over there."
In other deals:
— Boston sent right-hander Matt Albers and outfielder Scott Podsednik to Arizona for left-hander Craig Breslow and dealt utilityman Lars Anderson to Cleveland for right-hander Steven Wright.
— The Yankees traded right-hander Chad Qualls to Pittsburgh for infielder Casey McGehee, a move that followed injuries to Alex Rodriguez and Mark Teixeira.
AP Sports Writers Joe Kay, Stephen Hawkins, Rob Maaddi and Andrew Seligman, and AP freelance writers Alan Eskew, Dan Feldman and John Henry contributed to this report.Suggest a correction