Can antibiotics relieve an abscess without the need for surgery?

Surgery 101. Ideally all abscesses should be drained, and many will fail antibiotic treatment, because antibiotics can't circulate well inside the abscess. Superficial abscess will often come to head and drain on their own, but surgery speeds healing. Deeper abscess, inside the chest or abdomen, may be drained percutaneously with the help of a radiologist. Surgery verses waiting must be individually evaluated.
NotAllCreatedEqual. Treatment of an abscess depends on its cause & location, as well as the health of the patient. Some will resolve with antibiotics alone +/- (needle) aspiration, others will require surgical drainage +/- antibiotics. Regardless, all require medical attention and a search for the underlying cause.
Yes. An abscess can often be treated with antibiotics before moving to surgery but this would need to be monitored by your doctor. The size and location of the abscess is also important when deciding the proper management.
I would try it.... Sometimes an abscess will need to be drained for it to clear up, but a course of antibiotics is a good way to start. You can always have the surgery if things don't clear up - but make sure you keep your doctor posted as to any adverse changes in your health.