"If you still want to be stubborn about America's failure in Afghanistan, then remember the fate of Bush and [Pakistan's former president] Pervez Musharraf, and the fate of the Soviets and British before them," he added.
Al-Zawahiri also referred to Obama and other African-Americans who have served in US government in as "house negroes", an insult that comes from the time of slavery in the US, and used to describe a black person who does the bidding of white masters.
Obama's transition team has declined to comment on the tape, while Sean McCormack, spokesman for the US state department, dismissed the tape as just "more despicable comments from a terrorist".
"If anybody needed ... more of a contrast between what ... the West and the United States stand for, in terms of democracy and what these terrorists stand for, I don't think you need to go any further than those comments," he said.
In a major interview aired on Sunday, Obama reiterated his campaign promise to begin pulling troops out of Iraq and switch the military focus to Afghanistan.
Al Jazeera's Rob Reynolds in Washington DC said that if many people in the Arab and Muslim world were hoping for big changes in the US approach to dealing with al-Qaeda then they were likely to be disappointed.
Al-Zawahiri's message appeared chiefly aimed at convincing Muslims and Arabs that Obama does not represent a change in US policies.
Al-Zawahri said that Obama is "the direct opposite of honourable black Americans" such as Malcolm X, the 1960's African-American activist.
"America, the criminal, trespassing crusader, continues to be the same as ever, so we must continue to harm it, in order for it to come to its senses," al-Zawahri said.
In his speech, al-Zawahri refers to a November 5 US air raid in Afghanistan, meaning the video was made after that date.
Al-Zawahri said Obama's election has not changed American policies, which he said are aimed at oppressing Muslims and others.