Gaudin (11-11) struck out eight in six innings to end a three-start winless stretch in which he was 0-2, leaving the game with a 6-1 lead.
"That was big for Chad to go out there and have a great start," Swisher said.
Cust gave the A's a 2-0 lead in the first. Swisher hit his 20th home run, a shot to left-center in the third inning, after Daric Barton's leadoff double. Jack Hannahan added an RBI double and Marco Scutaro had a sacrifice fly in the fifth to give Oakland, which ended a three-game losing streak to the Rangers, a 6-0 lead.
Michael Young had three hits, including a two-run double in the Rangers' four-run seventh that pulled them within 6-5 in the opener of a four-game series between the bottom two clubs in the AL West.
"No one wants to finish last," Swisher said.
Young's two-out single in the third extended his career-best hitting streak to 13 games -- the longest this year by a Ranger.
"We were close, real close," Texas outfielder Marlon Byrd said. "We kept fighting, kept battling and have a chance to win this series. We're going to. We're coming out and going to win this series and finish up strong like we've been playing all year."
Gaudin, facing the Rangers for the second straight start, bounced back from allowing a season high-tying 10 hits Saturday in 3 2/3 innings -- which matched his shortest outing of the year. This time, he faced baserunners in every inning but was able to limit the damage after Jason Botts' leadoff single in the second.
"I felt positive toward the outing," he said. "I felt I was in control. I made some good pitches when I needed to and attacked the zone. I was not working under pressure, just laboring a little bit."
"Hey, if you want to be prime time, you want to be in the tight situations," Lugo said.
Kevin Millwood (9-12), still looking for his second straight 10-win season, allowed six runs and eight hits in 4 2/3 innings four days after he left Sunday's start against the A's having only allowed four hits because of a second rain delay.
"I just didn't throw good pitches," he said.
Millwood is winless in his last three starts.
"They just jumped on him," Texas manager Ron Washington said. "They got to him in the first, got to him in the third, got to him in the fifth. He got his pitch count up."
This series marks the final return of the season for Washington, the Rangers' first-year skipper who was beloved as a third-base coach in his 11 years with Oakland.
"Wash, we still love you!" one woman hollered from above the Rangers' dugout.
The A's improved to 72-76 and are determined to finish with a winning record for the ninth straight year. Only the New York Yankees (15) and Boston (10) have longer runs with above-.500 records.
The Rangers had scored 24 runs during their three-game winning streak over Oakland.
Gaudin has allowed 14 homers over his last 11 starts. ... Young is batting .556 (10-for-18) with a home run lifetime against Gaudin. ... Slugger Sammy Sosa was slated to start Friday night for Texas. ... Mark Ellis stole his ninth base in the third. ... The Rangers activated 3B Travis Metcalf from the 15-day DL after he had been sidelined since Aug. 29 with a strained left hamstring. He entered on defense in the eighth. ... David Murphy made the first of three scheduled starts in CF for the Rangers.