The England striker helped the hosts overcome a stuttering start when he lashed in an 18th goal of the season by beating Karl Darlow at the near post after 30 minutes.
Allan Saint-Maximin then scored a deserved equaliser with his first goal since coming back from a two-month absence caused by coronavirus symptoms, slamming in after Harry Maguire failed to clear.
The Newcastle winger threatened again early in the second half, but the game turned after 57 minutes when Daniel James re-established the lead.
That led to the visitors' confidence draining away, with Rashford then earning a penalty as he drew a foul from Joe Willock before Bruno Fernandes knocked in his 22nd goal of the season.
Defeat for Steve Bruce's side was an eighth in 10 Premier League games, leaving them three points above the drop zone - and a relegation scrap with Fulham is now set to ensue.
For Ole Gunnar Solskjaer's side, it was only a second win in six league matches, and a welcome victory after a draw at West Brom last weekend.
