Mengden, the young right-hander plagued by injuries this season, struck out eight over seven scoreless innings and the A's ended the season with their 17th win in 24 games, a 5-2 victory over the Texas Rangers on Sunday.
"A huge amount of confidence for me. For our team, I know this last month we did really well," Mengden said. "Come in here and do what I did, I really look forward to next year."
After winning the AL West title the past two seasons, the Rangers (78-84) slipped from 95 wins last year. They finished tied with Seattle for third place, 23 games behind division champion Houston.
Texas didn't have a base runner against Mengden (3-2) until Adrian Beltre's 3,048th career hit, a single leading off the fifth. Mengden, a second-year big leaguer who had foot surgery and a strained right oblique this season, walked one and allowed only four singles.
"Unbelievable, really," manager Bob Melvin said. "That's the best work we've seen from him."
Khris Davis hit his career-best 43rd homer for the Athletics (75-87), who finished at the bottom of the AL West for the third consecutive season, a franchise first. But they won six more games than last season, and Melvin got a contract extension this week, adding a year through 2019.
Beltre played for the first time since Tuesday, the night the Rangers were eliminated from wild-card contention. He missed only two weeks after a Grade 2 strain of his left hamstring Aug. 31, but was back two weeks later and was the DH in 13 consecutive games when the team still had a chance to make the playoffs.
After being lifted for a pinch-runner, Beltre got an extended ovation from the crowd that included a curtain call. The 38-year-old third baseman, who hit .312 in his 20th major league season, is signed with the Rangers through next season.
"It was nice," Beltre said of the crowd response. "I really appreciate that."
"For us to be able to take Adrian off the field like that and allow our fans to show their appreciation and for Adrian to show his love back to the fans, it was fitting," manager Jeff Banister said.
Blake Treinen worked the ninth for his 16th save in 21 chances.
Oakland went ahead to stay after four consecutive hits to open the third off Cole Hamels (11-6). Matt Chapman had an RBI double before Jed Lowrie had his 49th double, extending his single-season team record while driving in two runs.
Hamels struck out five in his three innings. The lefty, who missed more than seven weeks because of a right oblique strain, ended the season with 148 innings -- his fewest since 132 1/3 innings as a rookie with Philadelphia in 2006. He had thrown at least 200 innings each of the past seven seasons.
Davis, already only the Athletics player other than Jimmie Foxx (1932-34) with consecutive 40-homer seasons, went deep in the eighth. Davis hit 42 homers last season.
Elvis Andrus hit in the leadoff spot for the only time this season. He made starts in all nine spots in the batting order, the first Rangers player to do that in a season since Billy Sample in 1983. ... The Rangers didn't score until Nomar Mazara's two-run single in the eighth, giving him 101 RBI.
A's catcher Bruce Maxwell took a knee during the national anthem, something he did the last nine games of the season. He is the only Major League Baseball player who has done that, and the series at Texas was the first on the road. He kneeled in front of teammates before Sunday's finale, and loud boos were heard after the anthem.
Athletics: At home for their 2018 opener on March 29 against the Los Angeles Angels.
Rangers: Their 2018 opener is March 29 at home against Houston.
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