Morales led off the fourth inning with a homer and Zunino did the same in the fifth off Cleveland starter Ubaldo Jimenez, helping Seattle increase its league-leading total to 29 homers in July. The seven-game streak is the longest for Seattle since winning eight straight in August 2012.
Harang (5-8) gave up a solo homer in the first to Nick Swisher, and danced around trouble for his second win in the last three starts. Tom Wilhelmsen survived the ninth for his 22nd save.
Wilhelmsen blew a save in Cleveland earlier this season and gave up a one-out double to Michael Brantley. Carlos Santana grounded out, with Brantley advancing to third before Jason Giambi walked. Wilhelmsen got Lonnie Chisenhall to fly out to right for the final out.
Seattle was without manager Eric Wedge, who suffered dizziness during batting practice and had to be helped off the field and into the Mariners clubhouse by team personnel. Wedge was examined by team physicians Dr. Mitch Storey and Dr. Edward Khalfayan, and the team training staff at the stadium.
Despite Wedge eventually feeling better, the team decided to have him examined at a hospital. General manager Jack Zduriencik expected the skipper to be back at the stadium Tuesday. Beach coach Robby Thompson managed the team in Wedge's absence.
Cleveland pitchers had gone nine games 85 innings without allowing a home run. Jimenez (7-5) kept the streak going through 88 innings until Morales led off the fourth with his 15th homer of the season, which curled just inside the right-field foul pole.
An inning later, Zunino fell behind 0-2 and Jimenez tried to sneak a fastball past the rookie. But fastballs are the one pitch Zunino has shown he can hit at the major league level, and he lined a 411-foot shot into the left field bullpens for his second career homer.
Seattle has now homered in 25 of the last 26 games, the only break in the streak coming last Saturday when they won in Houston with just one hit.
Jimenez allowed only five hits and struck out six, but struggled with walks and a high pitch count. He walked four and was at 109 pitches when he was pulled with two outs in the sixth inning.
Swisher was hitting in the No. 2 spot for the first time this season. Indians manager Terry Francona hoped the switch might spark Swisher, who entered the night hitting .238.
The move worked early, as Swisher drove his 10th homer of the season out to left-center field in the first, then singled in his next at-bat in the third to put runners on the corners with one out. Harang escaped that jam with a strikeout of Jason Kipnis and a groundout by Asdrubal Cabrera. The Indians left Santana at third base in the fourth and Harang found a groove.
Harang retired 11 of the final 12 batters he faced. He threw only 11 pitches total in the sixth and seventh innings combined. He finished seven innings for the fourth time in 16 starts, allowed four hits and struck out two.
Notes: Seattle rookie SS Brad Miller was named the AL player of the week on Monday after hitting .385 with two homers and seven RBIs in Seattle's three-game sweep of Houston. He is the first Seattle rookie to be named player of the week since Ken Griffey Jr. in 1989. ... Swisher had hit in the No. 3 or No. 4 spot all season until Monday. ... Seattle RF Michael Saunders had a 10-game hitting streak snapped.