Note: in the picture below white space between the polygons is blank space – it does not contain any polygons as a result of Intersect or Union functions.
In ArcGIS, you should be able to use the Integrate tool to snap these features together (I am pretty sure it will work on a single feature class).
You should BE SURE TO MAKE A COPY OF YOUR DATA before running this tool, though, as the Integrate tool modifies the input data directly (you are not writing to a new output). You could however get around this limitation by creating a model that:
There are a number of caveats to this tool, and you should be sure to READ ALL OF THE HELP before performing this action on your data.
In ArcMap, if you are doing a limited number of features, you could edit the features using the Reshape Feature tool and the Trace tool.
First, start editing the layer and select the feature you want to snap to the other one. Then pick the Reshape Feature tool from the Editor toolbar. Choose a point inside the selected polygon to start, and then use the Trace tool (also on the Editor toolbar) to trace the edge you want to "snap" to. When you are done, you should end with a regular line segment back inside the polygon.
If the tracing proves difficult because your lines overlap, another option would be to use the Trace tool to draw new polygon with the boundaries you want. The Trace tool has an option to only trace selected features, so you can select the feature you want to trace the edge of.
If you only have a few of these to do, AND you are actually wanting to make the 3 features into one ... My suggested workflow is to do it in an edit session (in ArcMap). Just create a new feature and draw it like the one you drew in red +/-. Then select the new polygon and the other polygons that you want to merge and choose "Merge" in the Editor menu. If you have a lot to do, the previous answer (esp. the 2 step workflow) seems like a good way to go.