Dame Anne said the Government should lift the cap on contributions and allow transfers in and out, so employers do not opt for rival but potentially more expensive private schemes with fewer restrictions.
The introduction of auto-enrolment, which means employees have to actively opt out of contributing to a pension, began in October.
"Since auto-enrolment began, the case for lifting the restrictions has become even more powerful, and the need for action more pressing," Dame Anne said.
"For auto-enrolment to continue to work successfully, Nest must be allowed to thrive."
The Nest scheme is designed to be low-cost and simple to use. Thousands of companies are expected to make use of it over the coming six years as millions of private-sector workers begin saving for retirement for the first time.
The committee wants a relaxation of rules that cap contributions to a member's scheme at £4400 currently and prevent members from transferring money in and out to other schemes.
Dame Anne said: "The cap on annual contributions to Nest means that employers can't opt for Nest for their higher-earners or if they want to make more generous contributions. "
Groups including Age UK, Which?, the Federation of Small Businesses and the Trades Union Congress have already written to pensions minister, Steve Webb, calling for the restrictions to be relaxed.