Heavy snowfall caused a 13-minute delay to the start of the second half as officials, helped by St Etienne substitutes, cleaned the lines and painted them red.
Then play was stopped for 32 minutes in the 65th minute with St Etienne leading 3-1 after goals from Brazilian striker Brandao, Pierre-Emerick Aubameyang and Yoann Mollo.
The home side added a fourth goal, from Mathieu Bodmer, when the game resumed to climb to fourth place in the table on 40 points from 24 games.
St Etienne are two points behind third-placed Olympique Marseille, who occupy a Champions League qualifying spot and visit lowly Evian Thonon Gaillard on Sunday.
"We'll enjoy this win all the more...the conditions were very tough. It's special because of those who volunteered to sweep up the pitch even though they were not in the squad," St Etienne coach Christophe Galtier told a news conference.
Montpellier are ninth on 35 points.
"It's a shame. It was not possible to play football in these conditions," Montpellier coach Rene Girard told reporters.
"I shared my views with the referee but I'm just a coach. It was dangerous. I think some people have to take their responsibilities. There are problems in French soccer."
St Etienne took the lead after 13 minutes when Brandao powered the ball home from an Aubameyang pass following a fine team move.
Aubameyang made it 2-0 with a superb backheeled volley from a Renaud Cohad cross in the 29th minute.
With the two sides struggling with the snow-covered pitch after the restart, Mollo took advantage of the visitors' mistakes to give St Etienne a three-goal lead in the 48th, only to see Remy Cabella make it 3-1 a minute after.
The result moved St Etienne past Bordeaux, who were beaten 1-0 at lowly Ajaccio by a superb Ricardo Faty overhead kick in the 31st minute.
Bordeaux, who visit Dynamo Kiev on Thursday for their Europa League last-16 first leg, fell to sixth on 38 points, one behind Nice, who drew 1-1 at home with fellow European hopefuls Lorient.
The visitors deservedly went ahead in the 75th minute when left back Lucas Mareque skipped past centre back Renato Civelli's challenge to tip the ball over goalkeeper David Ospina.
Valentin Eysseric struck from outside the box seven minutes later to rescue a point for Nice.
Second-placed Lyon host 2011 champions Lille on Sunday, hoping to close the gap with leaders Paris St Germain to three points.