Try this.
Click Slide Show > Custom Animation.
Click Add effect > Entrance > Wipe.
Then right click on the effect, select Effect Options.
Click on the Chart Animation tab.
Click on the "Group chart:" dropdown, select By series.
Uncheck Animate grid and legend (if you do not want to animate it).
OK your way out.
Now, set the direction to "From Left".
See if this is what you want.

Select the chart on the slide. Right-click, choose custom animation. Add
effect|Entrance, choose wipe.

That effect will show up in the animation task pane. Double-click it in the
pane.

You'll be in a dialog box. Go to the Chart Animation tab. Choose "by series"
or "by category." Do not select the option to animate the grid and legend.

Then back in the animation pane, change the wipe direction to "from left."

If that doesn't give you as much control as you want, then you'll need to
ungroup the chart (twice), select the pieces of the line, group them, and
animate that line group.