Speaking at the Tasmania Economic and Political Overview event in Hobart, Mr Eslake said: "In my view there is no dispute that Tasmania experienced a recession in the 2012-13 financial year."
However, he said in the 2013-14 financial year Tasmania appears to be staging a recovery.
With the State election just a week away, Mr Eslake said whoever comes to power after the election will be presented with an economy that is picking up speed.
Mr Eslake said key areas the next government should look at include amalgamating smaller schools and public liability insurance.
He said the spend on education is greater in Tasmania per capita compared to mainland Australia, in part because its schools are smaller, but the results are well behind mainland Australia.
The results show that the longer students stay in school - often not long enough in Tasmania - the further they fall behind their counterparts on mainland Australia, he said.
On public liability insurance, Mr Eslake said Tasmania could offer all the opportunities that New Zealand has for adventure tourism but public liability insurance seemed to be what was holding back development of those opportunities.
He said the next government should look at reforms in this area.
Wells Economic Analysis, Principal and University of Tasmania (UTAS), Associate, Dr Graeme Wells said agriculture, minerals exports, tourism and the University of Tasmania were the "bright spots" for the economy.
"I think UTAS is going to have a significant effect on education tourism, if we think about tourism visitor nights a very significant increase in international students is underway at UTAS," he said.
"I don't see any recovery in forestry in the next 12 months, I don't think the uncertainty in terms of access to forests will be resolved in a way that gets investors back in the game in the next 12 months."
ANZ Chief Economist Australia, Ivan Colhoun
said for the last three years he'd been relatively negative about the Australian economy but "I think for the first time we have got a glass half full view of the Australian and Tasmanian economy".
In Tasmania he said in the last six months we've seen the whole economy improve, including house prices in Hobart and employment.
"When I look at job ads in Tasmania there is quite a lot of volatility…but the broad trend there is it is beginning to pick up and that is right across the state," he said.
"My message this year is more optimistic for both Australia and for Tasmania…we won't see a fantastically sharp turn around in growth but I think growth has begun to improve certainly in Tasmania and I think in Australia as well."
Speaking on the Tasmanian State Election, The Australian Tasmanian Correspondent, Matthew Denholm
said it was likely the Liberals would win a majority.
"I'd be surprised if the Liberals don't achieve the magic 13 seats needed to win a majority this time," he said.
"It appears likely Labor will be reduced to a rump of anywhere between six and eight seats. All the polling suggests the Greens will hold four seats of its current five and has a chance of holding all five," he said.
"The election is also a test for the new kid on the block the Palmer United Party (PUP).
"You would have to say if PUP can't get seats in Tasmania…you'd have to advise Clive to stick to dinosaurs and titanic replicas.