Fabinho tucked home a rebound to give Monaco a first-half lead at Roazhon Park, and defender Jemerson added a second from a corner shortly after the interval.
Brazilian full-back Jorge then netted his first Monaco goal in between a brace from Adama Diakhaby as Leonardo Jardim's side surpassed Bordeaux's single-season mark of 11 straight wins set during the 2008-09 campaign.
Monaco finished the season on 95 points, just one shy of the French record set by Paris Saint-Germain a year ago.
Ronny Rodelin atoned for missing a second-half penalty against PSG by grabbing a stoppage-time equaliser in a 1-1 draw at the Parc des Princes to ensure Caen retained their top-flight status.
Adrien Rabiot scored a 13th-minute opener and the visitors had looked to be heading towards a relegation play-off after Kevin Trapp kept out Rodelin's spot-kick.
Alexandre Lacazette hit his 100th Ligue 1 goal in his final appearance for Lyon with a brace in a 3-3 home draw against Nice.
The France striker, who is leaving his boyhood club this summer with Atletico Madrid touted as a possible destination, netted twice in the second half to finish with a career-best 28 league goals.
He was removed to a standing ovation before the finish, but Nice levelled at the death when Jean-Michael Seri converted a penalty deep into injury time at Parc OL.
Bafetimbi Gomis headed Marseille to a 1-0 win over Bastia to secure a return to the Europa League for Rudi Garcia's men, while defeat condemned Bastia to the drop.
- 'Goal of my life' -
Nicolas de Preville struck a hat-trick as Lille romped to a 3-0 victory over Nantes, while Dijon were grateful to a missed penalty from Martin Braithwaite as a 0-0 draw at Toulouse handed them the point needed to remain in Ligue 1.
Lorient will face Troyes, who finished third in the second division, in a two-legged play-off with their place in the top flight at stake following a 1-1 draw with Bordeaux.
Nancy dealt Saint-Etienne a 3-1 loss in Christophe Galtier's final game in charge of Les Verts, but the result was not enough to spare them from joining Bastia in Ligue 2 next season.
Strasbourg, who were demoted to the fifth tier after entering liquidation in 2011, returned to the top flight on Friday after sealing the second-division title with a 2-1 victory over Bourg en Bresse.
Amiens will play in Ligue 1 for the first time in the club's 116-year history after Emmanuel Bourgaud's 96th-minute winner at Reims bumped them up from sixth place to second.
"I see the ball coming, I shoot and it goes in. It's crazy, it's the goal of my career, the goal of my life," said man of the moment Bourgaud after his last-gasp heroics.
"We pushed and we pushed hoping for something like that to happen. This goal is extraordinary, it propels us into Ligue 1!" beamed Amiens coach Christophe Pelissier.
Third-place Troyes recovered from two goals down to beat Sochaux 3-2 but Bourgaud's goal with the final kick against Reims means they will face a play-off against Lorient.