Radamel Falcao's 78th-minute goal secured a 1-0 away win and moved the champions one clear of the previous best of 14 set by Bordeaux in 2009.
Monaco's run began back in February and has been completed without Kylian Mbappe, who was left out of Friday's squad amid rumours of an impending move away from Stade Louis II.
The visitors were made to work for their piece of history by a resilient Metz before Falcao's late intervention made it 23 league matches unbeaten.
"We're happy because we finished with a win and we set a new record of 15 wins in a row," Jardim said after the match.
"I think we played well even though in the first 30 minutes we missed a lot of passes.
"We accelerated in the second half, there was more mobility and intensity and that helped us win the match."
Monaco might have been forced to settle for a stalemate had it not been for new signing Rachid Ghezzal, who came off the bench to supply a superb pass for Falcao's winner.
The former Lyon man described his dream debut as a "beautiful evening", but knows he still has plenty to prove.
"This is a good first match [but] I have to keep working hard to earn my place," Ghezzal said.
"I don't come here with any guarantees."