Verdugo's two-out rocket to right off Shane Bieber (3-4) gave Boston a 3-2 lead in the sixth inning. Duran added an RBI single in the ninth off Enyel De Los Santos.
The scorching Red Sox moved a season-high 10 games over .500 and are 18-4 this month. Boston owns the second-best record in baseball since May 10 at 31-12, yet they remain 11 games back of the AL East-leading New York Yankees.
Right-hander Josh Winckowski (3-1) went 5 1/3 innings, allowing two runs, to win for the third time in three June starts. The Toledo, Ohio, native made his big league debut May 28 against Baltimore and was pitching for the first time in his home state.
Jake Diekman, John Schreiber and Matt Strahm followed Winckowski with 2 2/3 scoreless innings before Tanner Houck worked a perfect ninth for his sixth save.
Duran also stole two bases for Boston, which failed to score after loading the bases with none out in the eighth against Trevor Stephan.
The Guardians took a 2-0 advantage in the third when Amed Rosario tripled home Steven Kwan, then scored on Jose Ramirez's sacrifice fly. Ramirez leads the AL with 63 RBIs but had his 13-game hitting streak snapped.
Cleveland, which has won 17 of 24, fell to 0-2 on an 11-game homestand that continues against the Twins and Yankees. The Guardians entered the day tied with Minnesota atop the AL Central.
Bieber had dominated his previous five starts, going 2-0 with a 2.23 ERA and 38 strikeouts over 32 1/3 innings. The 2020 AL Cy Young winner lost for the first time since May 22.
Boston's Rob Refsnyder, who owns a .486 on-base percentage, did not play. He is the first player to reach base in his first 11 games with the team since Daniel Nava in 2021.
Red Sox RHP Connor Seabold is traveling on the taxi squad and will be activated Monday in Toronto. RHP Garrett Whitlock was an option to start against the Blue Jays but remains on the 15-day injured list with an inflamed right hip.
"Connor got his feet wet last season, so he knows how we do things,'' manager Alex Cora said. Seabold has made one MLB appearance, working three innings for Boston last Sept. 11 at the White Sox.
Red Sox: LHP Chris Sale (broken right rib) struck out six over 2 2/3 scoreless innings for the Florida Complex League Red Sox in his second rehab outing Saturday. Cora said the seven-time All-Star will next pitch at Double-A Portland.
Guardians: RHP Carlos Vargas (right elbow surgery), who has been on the 60-day IL since April 7, has made a pair of rehab appearances for Double-A Akron. The 22-year-old Vargas is 1-0 with a 0.00 ERA and two strikeouts in two innings.
Red Sox: LHP Rich Hill (3-4, 4.50 ERA) makes his final start of the month in the three-game series finale at Progressive Field. Hill has a 2-1 record with a 3.43 ERA in three June outings, striking out 21 over 21 innings.
Guardians: RHP Aaron Civale (2-3, 7.25 ERA) starts for the second time since spending a month on the IL with left gluteal soreness. Civale allowed two runs in five innings, taking a no-decision at Minnesota on June 21.