Rabbi Zelig Pliskin wrote:
Sincerely apologizing is the best way to overcome the resentment and negative feelings of someone who is angry with you. However, at times it can be very difficult to apologize. When you tell yourself, "Just apologize," it can become easier.
If you really did something wrong, you have an obligation to apologize. At times, it's wisest to apologize even if you aren't really at fault. Your apology can create harmony. Have the strength and courage to "just apologize." Your sincere apology will totally calm some people.
Even if you need to apologize a number of times and find the right things to say each time, your initial apology is still the first step forward.
(from Rabbi Zelig Pliskin's book: "Conversations With Yourself", p.135)