The Cubs lost for just the second time in 10 games following the All-Star break to fall a half-game behind idle first-place Milwaukee in the NL Central.
Engel lined his third homer of the season off reliever Justin Grimm (1-1) into the wind and several rows into the left-center bleachers to snap a 1-1 tie in the sixth. Davidson led off the eighth with his 19th homer, which sailed over the left-field stands and onto Waveland Avenue.
Miguel Gonzalez (5-9) allowed one run and seven hits in 7 1/3 innings to end a four-game losing streak and win for just the second time in 11 decisions.
Jose Abreu had two hits and doubled in a run for the AL-worst White Sox, who ended their longest slide in four years.
Monday's game was the first of four consecutive this week between the Cubs and White Sox.
In his first start since June 4 because of tendinitis in his pitching hand, Cubs right-hander Kyle Hendricks pitched 4 1/3 innings and allowed a run and eight hits. The 27-year-old was 16-8 with a major league-best 2.13 ERA last season and his return is one key to the defending World Series champions' rotation down the stretch.
Hendricks struck out five and walked none, but White Sox hitters battled to extend at-bats and drove up the righty's pitch count. Hendricks was replaced by Grimm with one out in the fifth after throwing his 92nd pitch -- one that resulted in Abreu's RBI double that tied it at 1.
White Sox reliever Anthony Swarzak got the final four outs for his first career save in eight chances. After Kris Bryant's infield single and a walk to Anthony Rizzo with two outs in the ninth, he struck out Willson Contreras looking.
The Cubs' John Jay had four hits, including a double.
Gonzalez's start was his second after missing nearly a month with right shoulder inflammation. He lost to Clayton Kershaw and the Los Angeles Dodgers last Tuesday despite allowing only one run in six innings.
Hendricks drove in the Cubs' first run on a broken-bat comebacker with the bases loaded in the second. The soft roller ticked off Gonzalez's glove, allowing Ben Zobrist to score from third, but second baseman Yoan Moncada threw out Hendricks at first.
The White Sox led 2-1 when Cubs loaded the bases against Gonzalez with two outs in the seventh. But Rizzo lined out to Engel on the warning track in center.
THE ROAD BACK
Hendricks made two rehab starts with Double-A Tennessee of the Southern League, allowing one run and two hits in 8 1/3 total innings. In his first start, Hendricks threw 45 pitches (27 strikes) over 3 1/3 innings. In his second, a week ago, the righty went five innings and 63 pitches (43 strikes).
To make room for Hendricks, the Cubs optioned RHP Felix Pena to Triple-A Iowa. Pe�a is 1-0 with a 4.50 ERA in 10 relief outings with Chicago.
White Sox: GM Rick Hahn said minor league RHP Zack Burdi will undergo Tommy John surgery this week. Burdi last pitched July 10 for Triple-A Charlotte.
White Sox: Left-hander Carlos Rodon (1-3, 5.75) makes his fifth start since returning from left biceps bursitis. The Cubs counter with righty John Lackey (6-9, 5.04) in the second game of the series, another matinee at Wrigley Field. Games 3 and 4 are at Guaranteed Rate Field and start at 7:10 p.m. local time.
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