After relief pitcher Greg Holland recorded two quick outs to start the inning in a 3-3 game, he allowed an Alberto Callaspo single and a Maicer Izturis double, then walked Peter Bourjos before Morales came to bat for Bobby Wilson and knocked the ultra-rare three-run single off the right-field wall.
Angels pinch-hitters had a Major League-worst .125 average (5-for-40) before the hit, which single-handedly doubled Morales' career RBI total as a pinch-hitter.
Morales had been batting .167 (6-for-36) in his last 10 games, but the clutch hit made the minor slump temporarily irrelevant.
"I think there's still some upside to Kendrys. I think he's getting comfortable," Angels manager Mike Scioscia said. "He's been through a stretch this month where he's just lost his comfort zone in the box, was chasing a lot of pitches. Hopefully this can start to get him back to where he is. We need him swinging the bat."
The Angels had taken a 3-2 lead into the seventh inning following a Maicer Izturis' RBI single in the sixth and had one of their two horses on the mound in C.J. Wilson.
But after a single from Eric Hosmer and a wild pitch to advance him to second, Wilson allowed a base hit to Brayan Pena on the left-hander's 113th and final pitch of the evening to tie the game.
In his first start since he allowed a season-high seven runs in six innings to the Yankees, Wilson lasted 6 2/3 innings, giving up three runs on nine hits and a walk.
Right-hander Kevin Jepsen came in and earned his first win of the season. After his 1 1/3 scoreless innings, Jepsen (1-1) has now allowed just four hits in his eight scoreless innings since being recalled on July 5.
Wilson has yet to earn a win in July, however, going 0-2 in his first five outings, again coming up short trying for his 10th win of the season.
"Tonight, I gave up three runs. I'm upset about that," Wilson said. "I'm upset about giving up as many hits as I did. So that's under my control. If I lose 2-1, then I can be frustrated. But if you get a no-decision because you're giving up three runs, that's your fault."
The Royals (40-55) took the early 2-0 lead on Wilson in the third inning on an RBI single from Alex Gordon and a Lorenzo Cain sacrifice fly to deep center. The fly ball could have been more, though, but Bourjos jumped and reached his glove over the fence, taking a potential home run away to limit the damage to just a single run.
The Angels would come back to tie the game in the bottom half of the inning on an RBI double from Howie Kendrick and an RBI groundout from Albert Pujols, who knocked home Mike Trout to extend the rookie's run-scoring streak to 15 games. Trout is the first Major Leaguer to score a run in 15 straight games since Matt Holliday with the Rockies in 2008.
When Bourjos is in the outfield along with Trout and Torii Hunter, he's part of a group that competes defensively with not only any outfield trio currently in the Majors, but may be one of the fastest, best ground-covering threesomes Scioscia has ever seen.
They're doing more than just limiting mistakes, Scioscia said, they're saving runs and contributing to wins more than most seem to notice.
"If you see it every day, you see the range, you see the guys that are taking hits away, helping our pitchers," Scioscia said. "Our defense is a plus, there's no doubt about that. I think there are a lot of things to great defense besides errors, we're making plays and also taking hits away."