The Blue Jays, pushing for a playoff spot, have won three straight and seven of nine.
''We know the type of team that we have,'' Toronto starter Tanner Roark said. ''We just keep clawing and scratching and fighting. We don't give up.''
Shaw went 3 for 4 with a double off the top of right field wall, Rowdy Tellez had two hits with a sacrifice fly and Randal Grichuk added a two-run single.
Yairo Munoz went 3 for 3 with a two-run homer and an RBI double for the Red Sox, who have lost five in a row, nine of 12 and fell a season-high 15 games under .500 for the first time since 2014.
The Blue Jays built an 8-4 edge before Boston rallied with three runs in the sixth. But Rafael Devers and Xander Bogaerts popped out with two runners on to end the threat.
''We played a good game today. We're just not getting that one big hit,'' Boston manager Ron Roenicke said. ''It seems like one big hit away a lot of times and then we pitch it differently at the end and it gets away from us. We did a great job of battling and coming back and the offense was better. Still a couple guys not swinging like we're used to, but hopefully we get them back swinging.''
A.J. Cole (2-0) pitched an inning of scoreless relief and Anthony Bass got three outs in the seventh and final inning for his fifth save.
Toronto will be the home team in the second game, which is the makeup of the Aug. 27 game in Buffalo, New York, that was postponed when the teams didn't play in protest over the shooting by police of Jacob Blake, a Black man, in Kenosha, Wisconsin.
This was the first time in Fenway Park's century-plus history that the Red Sox will be the ''road'' team, according to Elias Sports Bureau.
Entering the day, Boston's staff had allowed an MLB-worst 67 homers and owned the highest ERA at 6.13 - over a half run higher than the next worst, Colorado (5.51).
The Blue Jays didn't take long to add to those numbers when they made it 3-0 in the second inning against Zack Godley (0-4). Shaw's went into the seats behind Boston's bullpen and, after Joe Panik's walk, Jansen hit his into the Green Monster seats.
Roark gave up four runs and seven hits over 4 1/3 innings, striking out six with a walk. Godley allowed four runs in three with four Ks.
Blue Jays: Manager Charlie Montoyo said INF Bo Bichette (sprained right knee) was ''a lot better than thought he'd be at this time,'' and he'll hit in a simulated game against RHP Ken Giles on Saturday. . Montoyo also said Giles (strained right forearm) may do another simulated game or could be activated.
Red Sox: LF Andrew Benintendi (strained rib cage) has ramped up his workouts, but Roenicke said they're ''waiting to see from the medical end where he is'' as far as a possible return.
RHP Ross Stripling (3-1, 5.61 ERA) is slated to make his first start for Toronto in the nightcap. He was acquired from the Los Angeles Dodgers for two players to be named at the trading deadline. RHP Chris Mazza (0-1, 7.88) is set to go for Boston.
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