International observers led by the OSCE said the vote was skewed in favour of Putin during the campaign and the ballot was marred by irregularities.
"Overall, while the mission gives a positive assessment of voting on election day, it identifies problems with counting at some polling stations, unequal campaign conditions and limitations on voter choice," Hague said.
"These issues should not be overlooked... All allegations of electoral violations should now be thoroughly investigated."
In a statement, Hague said that a Russia with greater political freedoms, including the registration of political parties, freedom of assembly and freedom of the media "is in the interests of Russians and of the wider world".
Prime Minister David Cameron telephoned Putin on Monday evening and said he looked forward to working to overcome differences between the two nations.