Juventus was looking to preserve an impressive home record which had seen it lose just one of 16 European matches at the Juventus Stadium.
It was also boosted by Andrea Pirlo's return from injury after a three-week absence and the playmaker set up the first chance of the match with a delightful ball over the top which Morata volleyed over _ although the lineseman's flag was up for offside.
Carlos Tevez went close to scoring the opener in the seventh minute but Monaco goalkeeper Danijel Subasic dived smartly to his right to deny the Italian league's leading goalscorer.
Monaco appeared content to sit back and hit Juventus on the counter and it should have taken the lead in the 10th minute. Martial's pace took him past Leonardo Bonucci on the left flank and he picked out an unmarked Yannick Ferreira-Carrasco, only for the midfielder to fire straight at Gianluigi Buffon.
The Italy keeper was called into action again moments later to fingertip Ferreira-Carrasco's deflected effort around the left post.