Anthony Gordon opened the scoring early in the second half before captain Harry Kane added a second to seal the points for Thomas Tuchel’s side.
The visitors were handed a numerical advantage in the first half when Czech midfielder Pavel Šulc was shown a red card for a challenge on Elliot Anderson. Despite the extra man, England struggled to break down a stubborn defence before the interval.
The breakthrough arrived just two minutes after the restart. Trent Alexander-Arnold, making his first start under Tuchel, delivered an inviting cross from the right and Gordon stooped to head home for his third consecutive competitive goal for England.
Kane settled the contest in the 69th minute, sweeping home his 87th international goal after a deflection created the opening. The result gave England a first clean sheet in seven matches and moved them into second place in Group A3, three points behind world champions Spain.
Tuchel made several changes from the side that lost 3-2 to Spain at the weekend, including starts for Alexander-Arnold and Lewis Hall. The Three Lions limited the hosts to few clear chances after the red card and controlled large periods of the second half.
England will next face Croatia on Saturday as they look to build on the positive result and continue their recovery in the Nations League.