The 44-year-old man allegedly rammed vehicles with a stolen car in the city's south-east before arriving in the wealthy suburb of Brighton about 1pm on Monday where he threatened locals with the weapon and holed up in the house on Cluden St.
Police launched a large scale operation, which included the critical incident response team, and the man was eventually arrested about 3pm after being tasered was and taken to hospital where he remains under police guard.
The woman who owns the house on Cluden St, understood to be in her 60s, was not at home when the alleged break-in occurred on Monday afternoon and is not known to the man, police said.
The suburb is one of Melbourne's wealthiest and is home to a number of high profile residents including Bec Judd with her husband Chris and My Kitchen Rules star Zana Pali, who both have complained of rising crime rates.
Ms Pali, who was out trick-or-treating with her young family on Monday, shared a video of a police helicopter circling the suburb.