harry potter - How Were the Death Eaters Broken Out of Azkaban?

The key to this question is in your statement: "at the time of the break-out the Dementors were still under the control of the Ministry." Were they?

Remember, Dementors served whomever they felt like serving. They worked for the Ministry since they were guaranteed to continue their existence by feeding off the misery innate within Azkaban's walls. They are first and foremost Dark creatures, and they definitely were not bound to the Ministry. As Voldemort told his followers, the Dementors are his "natural allies."

Additionally, this wouldn't have been the first dereliction of duty from the Dementors that year. Earlier in the year, they had attacked Harry and Dudley in Little Whinging, as seen in Chapter 1 of Order of the Phoenix.

As Dumbledore said:

‘If it is true that the Dementors are taking orders only from the
Ministry of Magic, and it is also true that two Dementors attacked
Harry and his cousin a week ago, then it follows logically that
somebody at the Ministry might have ordered the attacks,’
said Dumbledore politely. ‘Of course, these particular Dementors may
have been outside Ministry control –’
‘There are no Dementors outside
Ministry control!’ snapped Fudge, who had turned brick red.

I believe they felt Voldemort's return, a "disturbance in the Force," so to speak. If they were not yet in contact with him, they probably knew he was back. They go where the misery goes, so to speak. They facilitated the DE escape from Azkaban, in order to set loose more misery elsewhere.

Hermione echoes this suspicion, when she reads the Ministry's claim that Black helped them escape:

‘I don’t believe this,’ snarled Harry, ‘Fudge is blaming the breakout
on Sirius?’
‘What other options does he have?’ said Hermione bitterly.
‘He can hardly say, “Sorry, everyone, Dumbledore
warned me this might happen, the Azkaban guards have joined Lord
Voldemort” – stop whimpering, Ron – “and now Voldemort’s worst
supporters have broken out, too.” I mean, he’s spent a good six months
telling everyone you and Dumbledore are liars, hasn’t he?’