That will enable the law to be changed in time for this coming Anzac Day, on April 25.
Mr Foster-Bell said that would act as an "ideal 100th birthday present for the RSA".
"It is a nice birthday present and, of course, it is then in place forever. Unlike the Rugby World Cup alcohol changes, this doesn't have a sunset clause."
He was pleased that the change had the support of all parties, and said Labour MP Kris Faafoi had been particularly helpful.
"It is one of those rare things that seems to have unanimous support across the whole House. That doesn't happen very often, and that's what allowed this to be fast tracked.
"Under the usual process it might have gone for months potentially, and it wouldn't have been done anywhere near in time for this year."
Green Party health spokesman Kevin Hague said his party - which took issue with Act leader David Seymour's successful push to allow bars to open for Rugby World Cup games - had no concerns about the RSA change.
"What Paul Foster-Bell's bill does is really just for a very, very small cross-section of licensed premises, and only for one day of the year. So it is a very minor extension.
"And I think absolutely nobody begrudges those old soldiers the opportunity for a drink once a year."
Mr Hague said the fast-tracked process was far from ideal, and was critical that the problem had not been picked up when the Sale and Supply of Alcohol Act 2012 was passed under former Justice Minister Judith Collins. Labour and NZ First were also critical that the 2012 legislation had missed the problem.
Current Justice Minister Amy Adams said there were no major flaws in the act.
"This [issue] with the RSAs, I looked at the issue and thought, actually, we do want RSAs to be open on Anzac morning. That is their biggest day of the year, and it makes sense to me."