The 28-year-old Russian will be provisionally suspended by the International Tennis Federation (ITF) from this weekend after she admitted at a news conference on Monday taking the banned substance meldonium.
Swiss watchmaker TAG Heuer cut ties with Sharapova this week, while Nike and Porsche have distanced themselves from her.
However, Eliasch said Sharapova, under contract with Head since 2011, had 'earned the benefit of the doubt' after making an 'honest' mistake.
"For a decade Maria Sharapova has been a role model and woman of integrity who has inspired millions of fans around the world to play and watch tennis," he said in a statement.
"The honesty and courage she displayed in announcing and acknowledging her mistake was admirable. HEAD is proud to stand behind Maria, now and into the future and we intend to extend her contract.
"We look forward to working with her and to announcing new sponsorships in the weeks and months ahead."
A statement issued later confirmed that the contract had been signed and extended.
In Los Angeles on Monday, Sharapova said she had been taking meldonium, also known as mildronate, for 10 years because of health problems and regular bouts of flu.
The drug, produced in Latvia but unavailable for purchase in the United States where Sharapova is based, was only added to the World Anti-Doping Agency (WADA) list of banned substances in January, shortly before her positive test in Melbourne.
Manufactured for people suffering from heart problems, it can also increase blood flow and improve exercise capacity.
Head said they were convinced Sharapova had not taken the drug to gain any competitive advantage.
"It is common ground within the scientific community that in order for meldonium to have any relevant performance enhancing effect it has to be taken in daily dosages in excess of 1,000 to 2,000mg," the statement said.
"According to the attorney for Maria, her dosage was significantly less than that."
While underlining Head's commitment to clean sport, Eliasch said he trusted Sharapova's explanation that she had not realised meldonium had been banned by WADA.
"In the absence of any evidence of any intent by Maria of enhancing her performance or trying to gain an unfair advantage through the use of mildronate, we further conclude this falls into the category of 'honest' mistakes," he said.
Russian biathlete Eduard Latypov was suspended from competition on Thursday after testing positive for meldonium, the latest in a flurry of athletes to fall foul of the drug since it was prohibited.
(Reporting by Martyn Herman; Editing by Angus MacSwan)
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