There are a few other reasons why Octavius has the last words. However, before I give these reasons, it would be good to give the the words of Shakespeare.
According to his virtue let us use him, With all respect and rites of burial. Within my tent his bones to-night shall lie, Most like a soldier, order'd honourably. So call the field to rest; and let's away, To part the glories of this happy day.
As mentioned above, Octavius is Caesar's adopted son. One of the characteristics that Caesar was known for was his clemency. So, to show clemency is to follow in the footsteps of Caesar. This shows continuity to the people.
Second, from a political point of view, this was wise, because the battle of Philippi was a civil war. So, to show mercy is to bring healing to Rome.
Finally, from a literary point of view, it was important that Octavius have the final words because he will be the first Roman emperor. So, it is fitting that he have the final words of honor. And if you recall, he would not give into Antony in the play. So, Octavius had to have the last words. This showed that he has won.
This speech is delivered after Antony has spoken about Brutus being the "most noble Roman of them all" since Brutus, alone, acted only for the good of Rome. All the other conspirators had some ulterior motive such as greed or jealousy, whereas Brutus was choosing the lesser of two evils...murder his good friend, Caesar, or allow Caesar to forever corrupt and ruin the Rome he loved.
Octavius speaks last, and also in favor of Brutus, proclaiming that Brutus will have an honorable burial. This is significant since Octavius is Julius Caesar's blood relative, and his forgiveness of Brutus is perhaps the most important. If Octavius can forgive Brutus for his loyalty to Rome which led him the murder Julius Caesar, then all of Rome should also be able to forgive and honor Brutus for his love of country.