Escobar had provided a 4-0 lead for the Royals in the second with his first career home run off starter Roenis Elias.
But the Mariners rallied behind Dustin Ackley's two homers and Kyle Seager's one to take a 7-5 lead in the fifth.
Kansas City took a 9-7 lead in the seventh, scoring four runs off reliever Danny Farquhar (1-1), highlighted by Giavotella's three-run blast over the left-field wall.
Entering the game, the Royals had hit just 14 home runs, lowest in baseball. But seven of their nine runs scored via the long ball.
Louis Coleman (1-0), who worked a scoreless sixth, earned the victory.
Greg Holland came on in the ninth to pick up his ninth save in 10 opportunities.
The Mariners committed a season-high five errors.
The Royals' seventh inning rally began with a double by Eric Hosmer. Farquhar then struck out Billy Butler and appeared to strike out Danny Valencia on a 2-2 curveball. But umpire Marcus Pattillo called it a ball, causing Farquhar to throw up his arms.
It seemed to unsettle the right-hander as he would walk Valencia and Alex Gordon, loading the bases. Lorenzo Cain tied the game with a sacrifice fly to center and Giavotella, promoted from Triple-A Omaha on Thursday, hit Farquhar's next pitch for his first home run.
A pair of singles and an error by third baseman Seager set up Escobar slam in the second. He hit Elias' first pitch, a 88-mph fastball, over the left-field wall. It gave the fifth-year shortstop his career high for RBI.
Ackley hit his first home run to open the third and Brad Miller, ending a 0-for-19 slide, added a two-run double for a 4-3 score.
A pair of errors in the fourth allowed the Royals to build the lead by to 5-3. Giavotella walked and Escobar singled. Catcher Mike Zunino tried to pick Giavotella off second but his throw scooted into center for an error, with the runners advancing. First baseman Justin Smoak then bobbled Nori Aoki's bouncer, allowing Giavotella to score.
Seager, who missed the previous two games with flu-like symptoms, hit his sixth home run in the fourth followed by Ackley's second, his first career multiple home-run game. Ackley also had a home run in the sixth inning Saturday, giving him three homers in three straight at-bats. He has four on the season.
The Mariners added a run in the fifth on Corey Hart's two-out single, scoring James Jones from second.
Seattle now turns its attention to Tampa Bay. The Mariners welcome the Rays for a three-game series starting Monday at Safeco Field.