The 44-year-old has spent the past two years serving as fast bowling coach under Bayliss.
Silverwood's first competitive series will be England's tour of New Zealand, which includes a five-match Twenty20 series starting in November and two Tests against the Black Caps.
While former India and South Africa coach Gary Kirsten and Surrey's director of cricket Alec Stewart emerged as contenders to succeed Australian Bayliss, managing director of England men's cricket Ashley Giles described Silverwood as the "standout candidate."
"I believe he is what we need to take our international teams forward," he said. "He is somebody we know well, but it is his intimate understanding of our structures and systems and his close relationships with Test captain Joe Root and white-ball captain Eoin Morgan that will help us develop our plans for the next few years."
"Ultimately, his highest quality is that he is a winner and that will be an important part of the job as we look to strive to become the most respected team in the world across all formats," he added.
The 44-year-old, who played six Tests and seven one-day internationals for England between 1996 and 2002, coached Essex to the County Championship title in 2017 before joining the England set-up.
Silverwood said he was excited to get started in the job.
"There is a tremendous amount of talent coming through, and there is enormous potential for growth," he said. "The hard work starts now, and I am confident we can make a positive impact during our winter tours of New Zealand and South Africa."
England won the 50-over World Cup in July for the first time but failed to wrest back the Ashes from Australia, who retained the urn after a 2-2 draw.