You can't do this well, at least as you have posed the problem. Conversely, you CAN do this, but you will do a poor job of it. For example, you COULD use the curvefitting toolbox, with a pure simple exponential model. That would lump any noise together with the oscillatory component. Unless the oscillatory component is perfectly orthogonal to the exponential decay component, then the decay will be measured incorrectly.
What you need to do to do this well is to choose a model for the oscillatory component, as well as the exponential component. Then fit the complete model together. Again, the CFT can do this, but only after you have chosen an intelligent model for the complete process.