The Labour leader's 34-hour deliberations - dubbed the “night of the blunt knives” by journalists - saw shadow ministers Michael Dugher and Pat McFadden sacked after they publicly criticised his policies.
Maria Eagle, who supports the UK's nuclear deterrent in spite of her leader's opposition, was also demoted from her role as shadow Defence Secretary.
Writing in the Observer, Mr Corbyn called on the party to show unity and oppose the “systematic undermining” of democracy by the Conservative government.
The leader said he was confident Labour could "build a coalition of electoral support that can beat the Tories in four years’ time".
Anger over the reshuffle saw three Labour frontbenchers - Jonathan Reynolds, Stephen Doughty and Kevan Jones - resign the following day in protest.
And the shadow Foreign Secretary Hilary Benn, though spared the sack, was reportedly told he could no longer directly disagree with Mr Corbyn from the front bench.
Mr Corbyn said he believes the reason Labour lost in May is because they failed to convince the public they offered “a real alternative”.
He criticised the Tory policies to cut public services and “[bank] on rising household debt to keep the economy afloat”.
And he said the Tories were involved in a power grab “to rig the political system to keep themselves in power and weaken the opposition, both inside and outside parliament”.
“The real middle Britain – of insecure self-employment, rip-off private pensions, unaffordable housing, mounting tuition fee debt and crisis-ridden social care – is crying out for a Labour government committed to fundamental reform," he wrote.
“My election as Labour leader represented a deep-seated desire for change and for a new direction in our politics. It’s hardly surprising if some find that shift difficult to adjust to.
“But we will maintain a broad-based leadership, with space for a wide range of views and all the talents.
“Labour stands for social justice and prosperity for all. It exists to deliver the jobs, security, services and life chances denied to so many. That is what Labour is for. That’s why we are building the genuine democratic alternative that the Tories are so determined to stifle”.
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