The principality side were 2-0 up and cruising when Sofiane Diop struck early in the second half to add to Krepin Diatta’s magnificent opening goal in Alsace.
However, Habib Diallo pulled one back for Strasbourg and then appeared to have salvaged a draw when he nodded into the net in the second minute of added time, only for the referee to disallow the goal for a marginal offside after consulting with the VAR.
It is a welcome win for Monaco before they travel to the Netherlands to face PSV Eindhoven in their Champions League third qualifying round, second leg in midweek.
Denied second place in Ligue 1 and automatic qualification for the Champions League group stage on a dramatic final night of last season, Monaco instead finished third and must therefore come through two qualifying ties in Europe’s elite club competition.
Belgian coach Philippe Clement made six changes to the side that drew 1-1 at home to Ruud van Nistelrooy’s PSV in the first leg, with Japan star Takumi Minamino — a recent arrival from Liverpool — one of those who dropped out.
Senegal international Diatta, who recently returned from a long-term injury, came in and opened the scoring in stunning fashion two minutes before the break at the Stade de la Meinau.
Axel Disasi nodded on a corner and Diatta connected first-time on the volley from the edge of the box to make it 1-0.
Monaco doubled their lead eight minutes after the restart when Diop converted from close range after Strasbourg goalkeeper Matz Sels could only palm out a low ball driven across the face of goal by Gelson Martins.
Strasbourg, who only narrowly missed out on European qualification themselves for this season, pulled one back in the 65th minute when Diatta’s compatriot Diallo rose to head in a cross by Thomas Delaine.
He then thought he had salvaged a draw when he nodded in after Alexander Djiku had helped on another Delaine cross, but the home celebrations were cut short by the VAR.
Reigning champions Paris Saint-Germain begin their defence of the title away at Clermont later on Saturday.
Kylian Mbappe will miss the game because of an adductor problem.
Lyon beat newly-promoted Ajaccio 2-1 on Friday in the season curtain-raiser, with Alexandre Lacazette scoring what proved to be the winner with a penalty in his first appearance since returning to the club from Arsenal.