The bottom club kept Monaco at bay after 11 consecutive defeats. United's 1930-31 team lost their first 12 games.
Monaco, who were without Brazilian Nene, the league's top scorer, and had South Korean forward Park Chu-young replaced at halftime through injury, dominated but proved too shy in the area.
Guy Lacombe's side dropped to fourth in the standings with 22 points from 12 matches, three points behind leaders Girondins Bordeaux who travel to Lille on Sunday.
"For us, it's like a victory. It's been a hard slog. Congratulations to everyone in the team," said Grenoble keeper Ronan Le Crom, whose superb save denied Yohan Mollo early in the game.
"We did not play beautiful football but for once, we don't go home with nothing in the bag."
Paris St Germain dropped to 12th after Loic Remy netted a last-gasp goal in Nice's 1-0 win at the Parc des Princes.
