Julio Borbon singled off Palmer to open the contest and was promptly thrown out trying to steal second. The Rangers went on to lose for the 12th time in 17 games in a late-season slump that took them out of playoff contention.
The Western Division-champion Angels won their fourth in a row and are set to meet Boston in the first round of the playoffs next week.
Los Angeles took the lead in the fourth on an RBI single by Torii Hunter, who came in to score on a base hit by Juan Rivera.
Most of the Los Angeles starters were replaced in the late innings.
Palmer (11-2) went five innings to get the win. Darren Oliver, Jose Arredondo, Kevin Jepsen and Rafael Rodriguez combined to complete the one-hitter.
The five Los Angeles pitchers walked five and the Rangers put another runner on base thanks to a ninth-inning error.
Derek Holland (8-13) gave up four runs on eight hits in 6 1/3 innings to take the loss.
Notable deaths of 2014 [PHOTOS]