Source: Sharecast
The decision marks "a significant milestone toward completing the transaction," it said.
The deal remains subject to regulatory approvals in Ireland and Austria.
Chief executive officer Mathias Döpfner said: "We are pleased to have received UK government approval to proceed with this acquisition. After a long period of uncertainty, we can confirm that we will invest significantly in The Telegraph’s editorial excellence and international growth."
Axel Springer announced last month that it was buying Telegraph Media Group, having scuppered rival takeover plans by the owner of the Daily Mail. It said the acquisition will strengthen its portfolio of premium journalism brands, which includes Politico, Business Insider, and Welt, among others.
Axel Springer first attempted to buy The Telegraph in 2004, but lost out to the Barclay family. It also bid unsuccessfully for the Financial Times in 2015.
Telegraph Media Group publishes The Daily Telegraph, The Sunday Telegraph and digital platform telegraph.co.uk.