Mills' opposition National Democratic Congress (NDC) said its own calculations, based on results from individual polling stations, put it past the 50 per cent threshold to avoid a second round.
Kwabena Adjei, the NDC chairman, said: "We are sure of a first-round win ... There are attempts to manipulate the result."
Amam Boateng, Al Jazeera's correspondent in Accra, said: "The problem with some of the votes being contested is that the two main parties have had their party agents at the polling stations.
"So they've been there, watching the count, and they've made their own numbers."
Adjei urged NDC supporters to remain calm "notwithstanding the attempt to steal the election".
'Good day fro Africa'
Kufuor, who marked his 70th birthday on Monday, steps down after serving the maximum two terms.
His successor takes over as Ghana looks to an era of greater prosperity when offshore oil becomes a significant factor of the economy in late 2010.
The elections are being keenly watched by African democracy activists after poll violence in Kenya, Zimbabwe and Nigeria.
|Elections are being watched closely after poll violence in Kenya, Zimbabwe and Nigeria [AFP]
The Electoral Commission and observers said turnout was high. Voting took place on Monday in some polling stations on Dwarf Island in Lake Volta after delays on Sunday.
"Polling was conducted in a calm and generally orderly manner ... EU observers assessed the overall environment positively," the EU team said in a preliminary report.
Valerie Amos, a British parliamentarian, who is leading an election monitoring team from the 53-nation Commonwealth, said: "I think it's a good day for Ghana and a good day for Africa generally, given some of the more recent examples of the difficulties we've seen in other countries."
The commission is due to announce the official result by the middle of the week, but the media in Ghana are permitted to announce provisional results as posted outside polling stations.
The election is the second handover of power through the ballot box since Ghana introduced multiparty democracy in 1992.
Kufuor won the 2004 election in the first round with 52 per cent to 45 for Mills. In 2000, he won power in a run-off, also against Mills, after former coup leader Jerry Rawlings stepped down.
In Sunday's parliamentary race, Akufo-Addo's NPP lost at least half a dozen seats to Mills's NDC, including that of Stephen Asamoah-Boateng, the information minister.
The NPP dominated the outgoing parliament with 128 of the 230 seats.
Samia Nkrumah, daughter of independence leader Kwame Nkrumah, won a seat in the family's home area of Jomoro.
Kufuor's centre-right administration has seen the economy grow by over 5 per cent annually in recent years and in 2007 launched black Africa's first Eurobond outside South Africa.
The former British colony is the world's second biggest cocoa grower and Africa's number two gold miner.
But many Ghanaians say the increased national wealth has passed them by, while critics say the government has done too little to combat widespread corruption and drug smuggling.