Watts, 58, stepped down on Wednesday as head of Europe's second-largest oil company with immediate effect and "by mutual consent", Shell said.
He will be replaced by vice chairman Jeroen van der Veer, 57, who is also president of Royal Dutch Petroleum, the Netherlands-based arm of the group.
Another senior executive, Walter van de Vijver, the head of Shell's exploration and production division, will also leave his post as part of a series of management changes.
The price of Shell shares surged on news of the board-room reshuffle, trading 2.1% higher at 383.75 pence in late deals. Royal Dutch shares rose 1.6% to 41.42 euros in Amsterdam.
"Royal Dutch/Shell has always been a company that doesn't like to react to shareholder pressure and reacts slowly," said Commerzbank analyst Clay Smith.
"For them to react this severely and this rapidly has got to be seen as a positive."
But Smith was doubtful whether the reshuffle would wipe the slate clean in the eyes of disgruntled investors.
"Management changes don't solve operational problems," he said.
The oil giant angered investors with its announcement on 9 January that it had re-categorised 3.9 billion barrels of oil and gas - one-fifth of its reserves - from proved to unproved.
The announcement sent Shell shares sliding by more than 10% and prompted calls from some investors for the chairman's resignation.
"Royal Dutch/Shell has always been a company that doesn't like to react to shareholder pressure and reacts slowly. For them to react this severely and this rapidly has got to be seen as a positive"
Watts was singled out for particular criticism because he failed to make himself available to investors to explain how the error over the reserves occurred.
He was also head of the exploration and production division when the disputed reserves were booked.
Watt's term was due to end in June 2005 when he reached the Shell's mandatory retirement age of 60.
But his position apparently became untenable as the backlash from the reserves debacle spread.
Last month the US Securities and Exchange Commission (SEC) launched a formal inquiry into the downgrade.
Shell is also facing a class action lawsuit in the United States from shareholders who allege the oil group deliberately violated accounting rules by misreporting its reserves in filings to the SEC.
Watts' departure is yet another coup for powerful institutional shareholders looking for a greater influence over the running over Britain's biggest companies.
It comes less than a month after supermarket chain Sainsbury was forced into a humiliating climb-down over the appointment of Ian Prosser, ex-head of the Bass group, as its future chairman, after strong opposition from big shareholders.