The bank also rebuffed a push by Moscow to end a freeze in investments in Russia over the Ukraine crisis.
The European Bank for Reconstruction and Development, at its annual meeting in Cyprus, also rebuffed a push by Moscow to end a freeze in investments in Russia over the Ukraine crisis.
One weak spot in the cautiously upbeat outlook is Turkey where growth is expected to slow as political uncertainty hits the economy, the EBRD said.
Average economic growth in the EBRD region -- which also includes countries in eastern and central Europe, central Asia and the Mediterranean -- is expected to pick up to 2.4 percent in 2017 and 2.8 percent in 2018, the EBRD said.
That would mark an improvement from growth of 1.8 percent seen in 2016.
"Overall the news is good. Growth is faster... despite all the risks, all the global challenges," EBRD chief economist Sergei Guriev told a news conference.
The rebound in world oil prices since the first quarter of 2016 has been positive for Russia as well as countries that rely on it for remittances and export demand, the bank said.
Russia is emerging from a two-year recession but even so the economic recovery is expected to be relatively muted with growth seen at 1.2 percent this year and 1.4 percent in 2018, the EBRD said.
"Even though recession in Russia is over, we forecast only a very slow recovery and we don't see remittances going back to the peak of 2013, which means for the countries that rely on remittances from Russia this is continuing to be a difficult time," Guriev said.
The London-based EBRD rejected a challenge by Moscow over the suspension of new investments in Russia that the development bank announced in 2014 as part of European action against Moscow over the Ukraine conflict.
"The board of governors overwhelmingly agreed that the bank has complied with its own internal rules with respect to its engagement with the Russian Federation," EBRD president Suma Chakrabarti said.
"That is a final and binding resolution."
Russian Economy Minister Maxim Oreshkin lashed out at the decision, calling it a "dangerous precedent".
Moscow has condemned the freeze as politically motivated and in breach of the mandate of the EBRD, which was founded in 1991 to help former Soviet bloc countries switch to free-market economies.
In its forecasts, the EBRD said growth in Turkey would slow further to 2.6 percent in 2017 from 2.9 percent in 2016 as security and political concerns hit investor sentiment and tourism.
"While increased disposable income on the back of a 30-percent hike in the minimum wage in January 2016 resulted in an increase in private consumption, growth was hit by a sharp fall in tourism receipts, Russian sanctions, and geopolitical tensions in the Middle East," the bank said in its economic update.
"Weak consumer and investor sentiment following the attempted military coup in July 2016 compounded these problems."
Growth in the southern and eastern Mediterranean region -- which includes Egypt, Jordan, Morocco and Tunisia -- is projected to pick up to 3.7 percent in 2017, from 3.4 percent last year.
The economies of southeast Europe are set to expand by around three percent on average both this year and in 2018, with Greece returning to growth as reforms progress and business confidence creeps back, the EBRD said.
The bank announced that its board of governors had given the green light for it to engage in the West Bank and the Gaza Strip for an initial period of five years to support the Palestinian territories with investments through a trust fund.