Morneau's pinch-hit highlighted a four-run rally for the Rockies, who scored in the ninth in each of the first two games of the series for the win.
Troy Tulowitzki had three hits, including a home run, and drove in two runs for the Rockies, who won extended their season-high winning streak to five games.
Madison Bumgarner pitched seven innings and homered in the fifth inning for the Giants, who have lost six of seven and were swept at home for the first time.
Matt Belisle (2-2) pitched an inning for Colorado. Juan Gutierrez (1-2) allowed four runs on three hits for the Giants.
LaTroy Hawkins pitched the ninth for his 14th save in 15 chances.
Tulowitzki singled to open the eighth and scored on Drew Stubbs' one-out triple. Michael McKenry beat out an infield single to score Stubbs and Charlie Culberson walked.
D.J. LeMahieu then hit a sharp grounder to shortstop Brandon Crawford, who bobbled the ball before getting the force at second. Ehire Adrianza also had trouble controlling the ball and his relay throw to first was late, setting up Morneau's hit.
The no-decision denied Bumgarner a chance to move into a tie for the NL wins lead with nine. He allowed four runs on six hits, walking four and striking out nine. Bumgarner had allowed a combined four runs in his previous 29 innings.
Rockies starter Juan Nicasio gave up three runs on four hits over two-plus innings. He walked four and struck out two.
Nicasio could not finish four innings in his second straight start, facing two hitters in the third before giving way to Franklin Morales. He's allowed 25 runs on 35 hits over his last 15 2-3 innings, a 14.36 ERA.
Nicasio walked Tyler Colvin and Crawford to open the second inning and Adrianza singled to put the Giants ahead. Bumgarner bunted and was safe when Nicasio threw the ball wide. Crawford scored on the play. Hunter Pence also drove in a run.
Tulowitzki's 18th home run of the season, a two-shot into the left field bleachers in the third, cut into the lead but the Giants answered when Hector Sanchez and Pablo Sandoval hit back-to-back home runs in the fourth.
Sanchez hit for Buster Posey, who left the game as a precautionary measure after taking a foul ball off his mask. Sanchez's previous 23 RBIs all came in Giants' victories.
Tulowitzki, the NL batting average leader, is hitting left-handers at a .441 clip (26 for 59) this season.
Brandon Barnes' RBI groundout in the fifth was matched by Colvin's RBI single in the sixth to make it 7-3 for the Giants.
Pinch-hitter Josh Rutledge added an RBI double against Bumgarner in the seventh to cut it to 7-4.
NOTES: Giants OF Angel Pagan was scratched from the starting lineup with lower back tightness. . The Giants will start RHP Matt Cain (1-4, 3.84) against the White Sox in Chicago on Tuesday. He is 8-10 with a 2.71 ERA in 22 career interleague games. . The Rockies will start LHP Tyler Matzek (1-0, 2.57) in their series opener against the Dodgers in Los Angeles on Monday. He'll be making his second career start. . Rockies' Wilin Rosario made his first start at first base. . Nicasio threw a season-high 38 pitches in the second inning . RHP Santiago Casilla is expected to be activated by the Giants in time for their series in Chicago. . Giants RHP Tim Lincecum turned 30. ... Giants manager Bruce Bochy was ejected from the game with two outs in the ninth for arguing balls and strikes.