Germany were in real danger of a group-stage exit from the tournament, with a draw only likely to give them a temporary reprieve, until Kroos curled in a 95th-minute free-kick to take the points.
Earlier, Mesut Ozil was dropped for Germany's must-win World Cup clash against Sweden in Sochi on Saturday.
Mexico lead Group F on six points after two games.
Neymar limped out of a training session with a sore right ankle following Brazil's opening 1-1 draw with Switzerland last weekend.
FIFA World Cup 2018, Highlights, Germany vs Sweden from Fisht Stadium, Sochi: Toni Kroos hit a fine freekick to give Germany a 2-1 win over Sweden
The victory leaves the Red Devils top of Group G with two wins ahead of their clash on Thursday against England.
Japan on Tuesday became the first ever Asian nation to beat South American opposition at a World Cup when they pipped Colombia 2-1.
Marcus Rashford will replace Sterling with Ruben Loftus-Cheek stepping in for Dele Alli, who suffered a thigh injury against Tunisia.
Falcao, Colombia's all-time top scorer with 29 goals in 74 appearances, has highlighted the danger of Bayern Munich's Robert Lewandowski, calling him "a striker who can settle matches any time".
FIFA World Cup 2018, Highlights Korea Republic vs Mexico: Mexico beat Korea Republic 2-01 to record their second successive win
The Asian giants' win against Colombia was one of the biggest surprises of the ongoing mega event and the Akira Nishino-coached side will be eager to dish out yet another spirited show.
Colombia face Poland in Kazan on Sunday, when a second defeat for Jose Pekerman's technically-gifted side would spell national disaster.
FIFA World Cup 2018, Highlights: Belgium recorded their second straight win by thrashing Tunisia 5-2.
Two-time Premier League Golden Boot winner, Harry Kane banished any remaining doubts over his ability to perform on the biggest stage for his country with the first double by an English player at the World Cup since Gary Lineker in 1990.
The win puts Switzerland in pole position to make it out of a tough Group E.
A largely frustrating afternoon for the five-time world champions was forgotten thanks to two goals in stoppage time.
Iceland got a lifeline in the 82nd minute with a VAR-awarded penalty but dead-ball specialist Gylfi Sigurdsson blazed over the bar.
Chelsea's Antonio Ruediger would be the likely replacement for Hummels.