The first bird emerged at the Cors Dyfi reserve near Machynlleth on Friday, followed by a second on Sunday.
The eggs were laid six weeks later than usual leading to concerns over whether the chicks would hatch.
Parents Monty and Glesni set up home on a 30ft (9m) tall man-made nest on 3 May.
They are one of only two known breeding pairs in Wales, with a second pair nesting near Croesor in the Glaslyn Valley.
Staff and volunteers at the Cors Dyfi nature reserve had an anxious wait for the chicks who took 32 hours to emerge from their shells.
Cracks in the first egg were spotted on Friday afternoon, with the bird finally hatching at about 21:00 BST.
The second chick started chipping away from inside its egg on Saturday afternoon but did not hatch until 20:40 BST on Sunday.
Monty and Glesni became a breeding pair for the first time this year after Monty's previous partner of two years, Nora, did not return from her African migration.
Ospreys return from migration in late March or April and usually lay eggs two or three weeks later.
But Monty and Glesni did not pair until 3 May and their two eggs were laid between 22 and 25 May - six weeks later than average.
Alwyn Evans, of Montgomeryshire Wildlife Trust, said: "By the time Monty finally paired off with his new partner, Glesni, a three-year-old bird from Rutland Water, it was getting late in the season.
"At that stage none of us knew if they would lay eggs, let alone whether they would hatch.
"It's been an emotional weekend to say the least."
Mr Evans said cracks in the second egg on Saturday were greeted with loud applause and shouting from visitors at the wildlife trust's osprey centre who had gathered to watch the birds hatch.
Last month the wildlife trust said ospreys were still persecuted and egg collecting remained a problem in the UK.
To protect the Powys eggs, more than 50 trust volunteers have worked 1,300 hours to ensure the nest has been watched 24 hours a day during the six-week incubation period.
The first osprey chick in the Dyfi valley for 400 years hatched in 2011.
Once prolific birds of prey in the UK, their numbers were drastically reduced in the 1840s after years of persecution, including egg collecting, hunting, taxidermy and loss of habitat.
The Dyfi Osprey Project has been popular with tourists, with an average of about 30,000 people visiting the nesting site annually.

Taylor, 30, started his career with the Latics, making 254 league appearances before joining Millwall in May 2012.
Former Bolton winger Obedayi, 27, scored two goals in 25 appearances for the Scottish Championship side.
He has also played under Oldham boss John Sheridan, having served under him at Chesterfield and Plymouth Argyle.
Find all the latest football transfers on our dedicated page or visit our Premier League tracker here.

Farmers in the north-eastern Liaoning province, made use of different colours and varieties of rice saplings in order to mimic the 3D effect.
The fields are housed as part of a theme park in Shenyang city, which also caters to weddings and camping trips for tourists.
The Xibo farmers, an ethnic Chinese group, have an annual tradition of creating patterns on their rice paddy fields.
Creating art on their rice fields also serves as a way of praying for blessings.
Last year, they produced 13 images featuring different themes of places, animals, and people.
Even the legendary Chinese deity Nezha was represented in rice paddy form.
The murals set a 2012 world record for the "largest rice paddy work ever made".
Officials are also hoping that this year's special 3D rice art will attract more tourists to Shenyang city.
"Stunning work," remarked a Facebook user. "Crop circles are so last century."
"It's Art Attack! In China," said another.

Sciver's single came from her first ball as England reached their target of 131 for the loss of six wickets.
A partnership of 75 between Charlotte Edwards (28) and Sarah Taylor (51) was the backbone of England's innings.
Katherine Brunt took 2-15 to help restrict New Zealand to 130-6, with Sophie Devine top scoring with 40.
Two osprey chicks have hatched at a nature reserve in Powys despite fears their parents had paired up too late in the mating season.
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Oldham have re-signed Bolton winger Chris Taylor on loan until the end of the season and brought in Tope Obedayi after his release by Dundee United.
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3D imagery has now reached China's rice paddy fields.
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