The ban was triggered because it was Cook's second such offence in 12 months.
England were ruled to be one over short of their target at the end of the rain-reduced game in Hambantota.
Cook, who will miss Sunday's fourth ODI in the seven-match series, was also fined 20% of his match fee.
Team-mates were docked 10% by the International Cricket Council.
England coach Peter Moores told BBC Test Match Special: "We knew Cook was on the verge of a ban. The bonus is that it wipes the slate clean before the World Cup."
Eoin Morgan will captain England, who trail 2-1, in Colombo.
England captain Alastair Cook has been given a one-match ban for his side's slow over rate in Wednesday's one-day international win over Sri Lanka.