And the 69-year-old could extend his stay further yet.
"The chairman has asked Archie and myself to continue in the job," said Brown, who took over as interim manager four weeks ago.
"We'll talk about the duration but certainly at least until the end of the season and hopefully, possibly, beyond. I think that will depend on results.
"The chairman was talking in those terms so we're pleased to hear that, and it will be okay for us to continue, we think.
"We are happy to accept the job and we'll do our best for Motherwell. There is a bit of work to be done but we have the nucleus of a good resource in terms of players. We enjoy working with the lads."
Well kept their unbeaten SPL record under Brown despite seeing midfielder Giles Coke sent off in first-half injury time for a foul on Scott Arfield.
Brown agreed with the decision but was happy with his side's work-rate after the dismissal.
The former Scotland boss said: "I don't think anyone would deny we deserved at least a point from the game and the lads were saying there was a penalty when (John) Sutton was barged off the ball."
Falkirk boss Eddie May, who lost Pedro Pele to an Achilles injury early in the game, did not think the red card was a huge help to his team.
"They just lost a centre-forward, they still had two banks of four," May said.
"You have to try and use your man but unfortunately our pitch is not great and not really for passing.
"So I don't think we got much advantage. You can't really have a touch going forward, because if you do it's usually at your knee.
"People will say you should really beat 10 men. I was at Ibrox on Saturday and Hearts went down to 10 men and scored a minute later. It's not easy playing against 10 men.
"I don't think they caused us any problems, even when they had 11 men."
Source: Team Talk