O'Shea told the LA Times: "[David and I] did not share a common vision for the future of the LA Times."
He became the third editor to leave the paper since 2005; his predecessors were also involved in budget disputes.
The newspaper said in a statement that it was "facing major challenges in charting a course that will be successful for the future. In that vein, we will be making several significant organisational changes to put us in the best position to succeed".
It said as a result of these changes, O'Shea would be leaving the newspaper.
O'Shea's departure comes one month after the paper's Chicago-based parent company, Tribune CO, was bought out in an $8.2bn deal led by US billionaire Sam Zell.
It is unclear when O'Shea, formerly a managing editor of the Chicago Tribune, will leave and who will replace him.
O'Shea had taken over from Dean Baquet, who left the Times in 2006 after a separate budget dispute. Baquet had succeeded John Carroll in July 2005 following his retirement. At the time, Carroll was reported to be concerned by continuing pressure to lower the newsroom budget.
The LA Times has a circulation of around 800,000 on weekdays and 1.2m on Sundays.
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