The Scottish world number three was visibly upset in his speech after winning the Brisbane International.
The dedication was directed at his close friend and Davis Cup team-mate Ross Hutchins, who is battling cancer.
Murray said: "I'd like to dedicate this victory to one of my best friends back home. You're going to be okay. You're going to get through."
When Murray signed a television lens after the match, he wrote "For You .....", the third word being indecipherable.
He was later asked about the dedication in his post-match press conference and replied: "Just a sick friend."
Murray and Hutchins, a professional doubles player from Wimbledon, have been close friends for a number of years and have played as doubles partners on several occasions.
Hutchins, 27, confirmed his illness on Twitter, writing: "Happy New Year to all. Unfortunately I will be away from tennis for a while as I was recently diagnosed with Hodgkins lymphoma."
He later added: "I am doing well, very positive with excellent medical support, friends and family around me. Looking forward to being back on the court soon."
The sportsman, whose regular doubles partner is Scot Colin Fleming, is ranked number 28 in the world.
Hutchins and Fleming won titles at Eastbourne and Delray Beach in 2012 and also teamed up for Team GB at London 2012. They won their first doubles title together in October 2011 and reached the quarter-finals at Wimbledon and the US Open in the same year.
In April 2011, Murray caused an uproar when he posted on Twitter that Hutchins was to be his new coach, claiming he "needed another yes man".
Several news outlets ran the story before it was flagged up as an April Fool prank.
Lawn Tennis Association chief executive Roger Draper said: "Everyone in British tennis is behind Ross. He is a true ambassador for the sport, both on and off the court, and I have no doubt he will overcome this setback. It's important that we give Ross and his family the privacy they need during this challenging time."
Hutchins is the son of former British Davis Cup captain Paul Hutchins and made his debut in the competition in 2008. In his absence, Fleming played with Jamie Murray at the Brisbane International this week.
Andy Murray defeated Bulgarian Grigor Dimitrov in a gripping final in Brisbane. The US Open champion played a flawless tiebreaker to win 7-0 and a clinical second set allowed him to recover a service break and complete ideal preparations for the Australian Open at Melbourne Park next Monday.
After his victory, Scotland's Sport Minister Shona Robison said: "Many congratulations to Andy who is taking his fantastic form of last season into 2013. The whole of Scotland will be behind him for the season ahead."
HODGKIN'S lymphoma is a cancer of the lymphatic system which affects more than 1600 people in the UK each year.
The cause of the disease remains unknown, although risk factors which can contribute to its development include poor immunity and previous exposure to the Epstein Barr virus, which causes glandular fever.
It can start in any part of the body, but the most common place for it to start is in the lymph nodes in the neck, followed by underarms, chest and groin.
In the UK, Hodgkin's lymphoma is most common in younger people, aged 20 to 34, and older people, aged 70 to 79, but it can occur at any age. It is slightly more common in men than women.
Treatment is usually very successful, even when it is in several different areas of the body. Most people can now be cured or the lymphoma can be controlled for many years.