Cookson's campaign was strongly supported by Makarov, the wealthy Russian cycling federation president, and he was asked to clarify his relationship with the oligarch.
"My relationship with Igor Makarov is the same as it is with every other member of the management committee and federation president," Cookson said. "He's an important figure in world cycling, there's no doubt about that. But I have made no commitment to Mr. Makarov, no funding from Mr. Makarov and there is nothing in any way out of order in our relationship."
After taking on the UCI role, the 62-year-old Cookson stepped down as president of British Cycling, which he led since 1997 and turned into one of the sport's most powerful national federations including Britain providing the last two winners of the Tour de France in Bradley Wiggins and Chris Froome.
Cookson's next order of business will be to call new IOC president Thomas Bach.
McQuaid will lose his IOC membership, and he warned during his final campaign speech that, if he were to lose the vote, it would hurt cycling within the Olympic movement.
"The truth is that the reputation of cycling doesn't depend on one man," Cookson said.
Another of Cookson's first moves will be to establish an independent anti-doping commission.
"There's no doubt that there's a lot of work to do and some tough decisions to be made," he said. "But with the benefit of a strong vote yesterday, a clear message from world cycling that they want a change in leadership, I think I've got the support and authority from all my friends and colleagues around the world to make a new start."
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Follow Andrew Dampf at http://twitter.com/AndrewDampfAP