Hewitt, 33, produced some inspired tennis to race through the opening two sets on Rod Laver Arena against a player who had never won a five-set match in his career.
However, the former Wimbledon champion was unable to maintain that level and Becker – who admitted to kicking a door to let off some steam after leaving the court following the second set – roared back to secure a 2-6 1-6 6-3 6-4 6-2 victory in just over three hours.
Becker’s win over Andre Agassi in the third round of the US Open in 2006 prompted Agassi to retire, but the 33-year-old German said: “I hope it’s not Lleyton’s last match because then I get asked about it all the time. He is a great warrior, an unbelievable player, so I hope he continues.”
Asked about his future plans, Hewitt – who won just one grand slam match in 2014 – said: “I’ll sit down and think about it. As I’ve said the whole time, I haven’t thought about anything.
“But obviously the Davis Cup is the next main thing. Now that we’ve got some guys playing really good tennis at the moment, it’s an exciting time. We have a good chance to possibly pull off an upset away (against the Czech Republic in Ostrava). That’s the next focus.”
Top seed and four-time champion Novak Djokovic lost just five games as he cruised into the third round.
World No 1 Djokovic thrashed Russian Andrey Kuznetsov 6-0 6-1 6-4 in an hour and 24 minutes.
Djokovic has dropped just two sets over the first and second rounds since 2007, winning his first title at Melbourne Park in 2008 and lifting the trophy three years in succession from 2011.
Defending champion Stan Wawrinka, who ended Djokovic’s 25-match unbeaten run here in the quarter-finals last year, also safely reached the last 32, but needed two hours and 16 minutes to finally subdue the resistance of Romanian qualifier Marius Copil.
The first two sets took longer to complete than Djokovic’s entire match as they lasted an hour and 35 minutes, with Wawrinka winning both in tie-breaks before taking the third set 6-3.
Wawrinka will face Jarkko Nieminen in the third round after the Finn defeated German qualifier Matthias Bachinger in straight sets, with fifth seed Kei Nishikori taking on American Steve Johnson after coming from a set down to beat Croatia’s Ivan Dodig.
Top women’s seed Serena Williams reached the third round for the 14th time in 15 appearances, but had to save three set points when trailing 5-3 in the opening set before reeling off 10 games in a row to beat Vera Zvonareva 7-5 6-0.
Venus Williams, who beat American compatriot Lauren Davis 6-2 6-3, will play Italy’s Camila Giorgi in the third round and is scheduled to meet sixth seed Agnieszka Radwanska in the last 16. Radwanska thrashed Sweden’s Johanna Larsson 6-0 6-1. Fourth seed Petra Kvitova beat Germany’s Mona Barthel 6-2 6-4, but eighth seed Caroline Wozniacki fell to two-time champion but unseeded Victoria Azarenka 6-4 6-2.