The surging Mariners won their fifth straight and 10th of 13 after an 8-19 start.
Seager lined his sixth home run of the season in the first inning off Texas starter Jordan Lyles (1-4). Lewis added his ninth of the season leading off the fourth to deep center field.
"It's fun. It's fun to watch it every day," Seattle manager Scott Servais said. "You can score runs in a number of ways. You can prevent runs in a number of ways. It's just not hit the ball over the fence and hope you strike everybody out. Everybody's contributing."
Seattle rookie Justin Dunn continued his impressive recent run, throwing six innings and allowing two runs on four hits. Dunn (3-1) walked four batters in the first three innings but recovered to pitch through the sixth for the third consecutive start. Since lasting just two innings and allowing six earned runs against the Los Angeles Dodgers on Aug. 17, Dunn has given up two earned runs over his past 18 innings.
Dunn said the difference of late is going into games with a solid game plan of how to approach hitters.
"(I'm) now going into games understanding how I'm trying to set each guy up," Dunn said. "Each pitch has a purpose and I'm able to throw with a lot of intent and conviction, and let my stuff play."
Seattle cut down the potential tying run at the plate to end the seventh inning. Taveras lined a shot into the left-center field gap. Lewis cut the ball off before it reached the wall. J.P. Crawford made a strong relay throw and catcher Joseph Odom tagged Derek Dietrich, who was trying to score from first base.
"Those guys in the dugout, they were about as excited that they executed that play as they were anytime seeing those guys hit a home run," Servais said.
"Today we got some walks and stayed in the strike zone and we hit it hard," Texas manager Chris Woodward said. "This year for some reason, we're having a hard time finding holes. We're still applying pressure, which we need to keep doing."
Lyles lost his third straight decision despite pitching a season-high 6 1/3 innings. He allowed three hits, struck out four and walked four.
Gallo joined a select few who have homered into the upper deck of right field with his second-inning blast. Mo Vaughn, Carlos Delgado, Nomar Mazara and Daniel Vogelbach are the only players to have previously homered into that section of the stadium since it opened in 1999.
"That was one of the first homers I've seen that went that far off a lefties bat and stayed that true to get up there. That was impressive," Dunn said.
Seager had two hits to move past Jay Buhner into fourth place on Seattle's all-time hit list. Seager has 1,256 hits in 1,301 games.
Texas placed LHP Joely Rodriguez 10-day injured list with a left hamstring strain and activated LHP Jesse Chavez to take the open roster spot. Rodriguez was injured during Saturday's loss to Seattle and left the game in the middle of an at-bat. Texas manager Chris Woodward said initial exams have him skeptical Rodriguez will be able to return before the end of the regular season.
Rangers: LHP Kolby Allard (0-4, 5.40) makes his seventh start. Allard has allowed 24 earned runs in 16 innings over his past four starts combined.
Mariners: LHP Marco Gonzales (4-2, 3.09) pitches in the series finale. Gonzales threw the second complete game of his career in his last start, a 2-1 win over the Angels.
More AP MLB: https://apnews.com/MLB and https://twitter.com/AP-Sports
Former President Barack Obama gives shoutout to Tampa Bay Rays at Florida rally
Former President Barack Obama spoke at a rally in Orlando, Florida, on Tuesday in support of Democratic presidential candidate Joe Biden and had some encouraging words for the Tampa Bay Rays ahead of Game 6 of the World Series.
Game 3 between Los Angeles Dodgers, Tampa Bay Rays least-watched World Series game ever
The Dodgers' 6-2 Game 3 win over the Rays on Friday was the least-watched World Series game ever, beating the record set two days earlier in Game 2. Friday night's game was seen by an average of 8,156,000 viewers
World Series Daily: Los Angeles Dodgers eye first title in 32 years as Tampa Bay Rays try to force Game 7
Tampa Bay sends Blake Snell to the hill Tuesday night, while L.A. counters with a bullpen game. Will the Commissioner's Trophy make an appearance?
World Series 2020: What Tampa Bay Rays, Los Angeles Dodgers need to do from here
After a day off before the final game -- or games -- we map out the path to victory for both teams.
World Series 2020: Our impressions from inside Globe Life Field
Our reporters have been covering the Fall Classic from inside MLB's newest ballpark. Here's what they say about a stadium few have seen in person.
Doug Mathis, Brendan Sagara to share pitching coach duties for Rangers
The Texas Rangers plan to utilize Doug Mathis and Brendan Sagara as co-pitching coaches in 2021.