It was a superb first-half performance from the Republic of Ireland, who put France under pressure from the early stages.
Megan Connolly's free-kick forced Constance Picaud into a strong save, and Anna Patten should have opened the scoring when she headed straight at the French goalkeeper from the resulting corner.
Murphy also had a great opportunity for an early opener when she brought down Aoife Mannion's long ball with a superb touch but her finish lacked power and Picaud easily saved.
McCabe's opener was well deserved as she won possession in her own half and found Murphy, who raced forwards with the ball.
The Irish captain continued her run into the area and Murphy found her with a neat reverse pass, and McCabe fired low into the net to spark wild celebrations.
At the other end, France were limited going forward as Clara Meto and Sakina Karchaoui fired well wide in the visitor's best chances.
The pattern continued after the restart as the Irish defence stifled France, who were limited to half opportunities, including an effort from Sandy Baltimore that flew way over the bar.
Patten's header from a McCabe free-kick was clawed away by Picaud, and Denise O'Sullivan nearly scored a superb goal as her flicked effort fell just short of beating the French goalkeeper.
But the French for the equalise when Malard danced inside McCabe and slotted into the bottom corner past Courtney Brosnan.
The Rpeublic responded well and Murphy almost got on the end of McCabe's ball into the area, and at the other end Chloe Mustaki had to be alert to cut out 's dangerous cross.
However, Malard struck the decisive goal when the Republic failed to deal with a corner
O'Sullivan almost grabbed a dramatic equaliser when she fired towards goal after a goalmouth scramble but substitute Katoto headed off the line, before she was mobbed by her team-mates for the intervention.
