He made the remarks in a telephone conversation with Deputy Prime Minister of Azerbaijan Shahin Mustafayev late on Wednesday, IRNA reported.
Vaezi said Tehran welcomes Baku's proposal for a more active and prominent role by Iran, Russia and Turkey in maintaining peace and stability in the region.
Renewed fighting over Nagorno-Karabakh – an ethnic Armenian region of Azerbaijan that broke from Baku's control in a devastating war in the early 1990s – has claimed some 400 lives and forced thousands of people from their homes.
The heavy clashes erupted late last month, with both sides blaming the other for the biggest outbreak in violence since a 1994 ceasefire left the status of Nagorno-Karabakh in limbo.
In a bid to put an end to the escalating conflicts between the two warring sides, Moscow hosted a peace talks on Friday with the presidents of Azerbaijan and Armenia to broker a ceasefire.
Welcoming the ceasefire, Vaezi called on the governments of the Republic of Azerbaijan and Armenia to respect ceasefire and resolve their differences through dialogue and within the framework of international law and respect for the recognized borders and territorial integrity of the two countries.
Vaezi also expressed satisfaction with the implementation of the agreements between Iran and Azerbaijan, and especially the recent agreements presented in a telephone conversation between the two presidents, saying that the realization of these agreements will undoubtedly be an effective step in the process of expanding bilateral relations.
Mustafayev, for his part, said that his country is keen on promoting cooperation between Tehran and Baku in all areas of mutual interest.
He also presented a report on the conflict, saying that Baku favors Iran, Russia and Turkey to play a more active role in addressing regional problems.
Agreements between the two countries are being implemented and pursued rapidly, as the implementation of these agreements will be an effective step in line with developing joint cooperation, he said.