The Angels have scored only 56 runs in their last 18 games -- no more than five in any of them.
"Eventually, we've got to start hitting the ball," manager Mike Scioscia said. "We believe that will happen and the hits will start coming."
Detroit took the lead in the second game on Monroe's fourth homer and went up 3-0 in the fifth inning when Inge connected off Jarrod Washburn (2-2) for the second time in six days.
Inge also homered off Washburn in a 10-1 Tigers rout in Anaheim last Sunday.
"We were really frustrated after the first game, so it was good to get some runs early in the second one," Inge said. The Angels won the opener 4-2.
The Angels scored a run in the sixth when Darin Erstad tripled and scored on Orlando Cabrera's groundout.
Los Angeles added another run in the seventh on Jose Molina's RBI single, but Franklyn German came out of the bullpen to retire pinch-hitter Vladimir Guerrero and Chone Figgins.
Mike Maroth (4-2) improved to 4-1 in his last six starts, giving up two runs and seven hits in 6 1/3 innings. Four Tigers relievers finished, with Ugueth Urbina pitching a perfect ninth for his fourth save.
"I just came out and threw strikes," Maroth said. "This was a total team effort -- the defense played well, we got some hits and the bullpen was great."
Scioscia came out to argue with plate umpire Hunter Wendelstedt after the last two batters -- Steve Finley and Dallas McPherson -- were called out on strikes.
"The pitches weren't the problem," Scioscia said. "Steve and Mac didn't like the calls, and they said something, but I took exception to his attitude of stalking them back to the dugout. Let them have their say and go on."
After rain postponed Friday night's game, the start of Saturday's doubleheader was delayed 26 minutes by showers -- and there was a 39-minute delay in the sixth inning. But Game 2 was played under sunny skies.
It was the Angels' first doubleheader since Sept. 6, 2003, in Kansas City. ... Umpire Chad Fairchild made his major-league debut, working at third base in the first game. ... Doug Creek made his Tigers debut in the opener, pitching in the majors for the first time since 2003. ... Chris Prieto, 32, made his major-league debut in the second game, starting in center for the Angels. He went 0-for-1.