They were put on the market in March and bought by the Sedgwick family, who bought the town's North Pier in 2011.
Present owners Crown Entertainment Centres sold the pair of piers to Peter Sedgwick, who currently operates rides on both piers.
Central Pier - which is 341m (1,118ft) long - houses a distinctive 33m high (108ft) Ferris wheel.
The Central Pier and Blackpool South Pier, which is 150m (492ft) long, do not benefit from listed status, despite opening in 1864 and 1892 respectively.
At the time the sale was announced, a spokesman for the National Piers Society - which promotes the preservation and continued enjoyment of seaside piers in the UK - described them as "in good order and trading successfully".
Richard Baldwin, director at Bilfinger GVA Retail, Hotels and Leisure, which completed the sale on behalf of the leisure operator, said: "After considerable interest in both piers, we are delighted to have completed these two sales in such a short timescale.
"The conclusion of these sales... is a clear illustration of a strengthening in the alternative leisure markets, which is extremely encouraging."
St John Stott, director at Crown Entertainment Centres, said: "Peter Sedgwick is an experienced, longstanding leisure operator and we are delighted to have sold both the Central and South Piers to him."
The North Pier was badly damaged in storms in 2013 which cost about £1m to repair.
Mr Sedgwick said: "To add these two iconic piers to our portfolio is extremely exciting and we hope to capitalise on a group ownership of all three Blackpool Piers to ensure that many thousands of visitors continue to enjoy these wonderful attractions for years to come."
Blackpool's Central and South piers have been sold for about £8m to a family with close ties to the town.