Hawkins, 41, who could return next season on a $2.25 million club option, is two appearances behind Hall of Famer Goose Gossage, in 15th place.
"I had a lot of managers that had a lot of faith in me, a lot of good teammates, a lot of good mentors in the game, and plenty of blessings from the man upstairs to stay healthy," Hawkins said.
Rockies reliever Matt Belisle pitched scoreless ball in the 10th and 11th. But the Rockies (66-95) -- who must win Sunday's regular-season finale to avoid finishing with the second-worst record in club history -- would fall to a Dodgers squad that has clinched the National League West.
Morales (6-9) gave up a Scott Van Slyke single and hit Tim Federowicz. Van Slyke stole third, and scored on a Morales wild pitch. It made a winner of righty Carlos Frias (1-1), who threw three scoreless and hitless innings.
Early, right-handed prospect Eddie Butler showed some of the promise that made Rockies fans anticipate his arrival -- before the Dodgers delivered harsh lessons, to the tune of five runs and eight hits in 4 2/3 innings.
Butler, 23, a supplemental first-round pick in 2012 out of Radford University, gave up Adrian Gonzalez's 26th homer of the season for a 1-0 deficit in the first. But he stayed just ahead of hitters and, thanks to 2010 No. 1 pick Kyle Parker's first career RBI on a double in the fourth and Rafael Ynoa's RBI single in the fifth, he carried a 2-1 lead into the bottom of the fifth against the Dodgers and starter Dan Haren.
"My two-seamer worked early, but somewhere about the third inning it started going away -- I started losing the feel for everything," Butler said.
In the Dodger fifth, Yasiel Puig reached on a one-out fielder's choice, stole second and took third on catcher Michael McKenry's wild throw. Van Slyke singled through a drawn-in infield, and Matt Kemp added an RBI double. Andre Ethier would later add an RBI double. A.J. Ellis added an RBI single off Juan Nicasio for a 5-2 Dodgers lead.
Butler made his Major League debut against the Dodgers in a loss on June 6, but suffered a right rotator cuff strain in that game. After recovery and rehab he finished up in the Minors and two more Major League starts. But he will go to the Arizona Fall League next week to continue pitching to regain his form.
McKenry and Parker singled with one out in the sixth off Jamey Wright, and Charlie Culberson's RBI single cut the difference to two runs. McKenry opened the eighth with his eighth home run of the season, off Yimi Garcia.