Barkley brought the 2-1 advantage to England after picking up a deflected shot in the 39th minute.
In the 59th minute, Barkley scored another goal from some 12 meters, while Harry Kane increased the advantage to 4-1 in the 71st minute on Raheem Sterling's assistance.
The final score was set in the 81st minute after Sterling caught through ball of Jordan Henderson and slipped it between goalkeeper's legs.
World champions France met little resistance from Iceland en route to a 4-0 home win as Paris Saint-Germain's striker Kylian Mbappe contributed one goal and two assists.
Samuel Umtiti headed the opener for Les Bleus. Olivier Giroud bagged his 35th goal to sit third among France's all-time leading scorers. Antoine Griezmann completed the rout at the Stade de France.
France sit atop Group H with a superior goal difference over Turkey, who thrashed Moldova 4-0 at home.
Albania registered a 3-0 win away to Andorra in the other match of the group.
