The average UK house price stood at £175,826 in December, up 8.4% annually but still 5% down on the 2007 peak.
The value of homes across the country rose by 1.4% on average in December, up from £174,566 in November, according to the Nationwide house price index.
The building society's quarterly breakdown of the data revealed that house price growth in London hit 14.9% in the last three months of 2013. It said that was the highest growth rate since the first quarter of 2010.
The price of a typical London home stood at £345,186 by the end of the Q4 2013, or 14% above their 2007 peak.
The data also revealed house price recovery is spreading across many UK regions. For example, prices in Northern Ireland were up 7% year-on-year - although Nationwide pointed out they had risen from a low base and are still "around half the level prevailing in late 2007".
Prices in Scotland saw a 3.7% annual increase during the last quarter of 2013, while in Wales they rose to 6.1% from 3.6% in Q3.
Meanwhile, in England prices in the South and the Midlands continued to outperform the North, which was the weakest English and UK region for house price growth – at just 1.9% over the year.
Robert Gardner, Nationwide's chief economist, said: "The UK housing market followed the trajectory of the wider economy through 2013, gaining momentum as the year progressed. The average monthly increase in house prices rose from 0.4% in the first half of the year to 1% in the second half of 2013."
He added: "The upturn also became increasingly broad based over the course of 2013. For the second successive quarter, all thirteen UK regions saw positive annual house price growth in Q4, though London and the South East continued to record the strongest pace of growth."
He explained that "further improvements in the labour market and the brighter economic outlook" had helped to bolster sentiment among potential buyers.
However, he added: "Part of the reason for the acceleration in house price growth is that the supply side of the market has not kept pace with the upturn in demand, even though buyer numbers remain subdued by historic standards.
"For example, in Q3 2013 the number of housing transactions in England was around 25% below pre-crisis levels, while the number of new homes built was around 45% lower."