Manchester United climbed back into the Premier League top four with a 3-2 win over Antonio Conte's side at Old Trafford, with Ronaldo scoring all three goals.
The 37-year-old took his tally for the season to 12 goals in 24 league games.
Ronaldo scored his second Premier League hat-trick and first since January 2008 against Newcastle. His two trebles were separated by 14 years and 59 days, the longest gap between hat-tricks for a player in Premier League history.
Outspoken former United captain Keane was impressed with Ronaldo's performance and hopes the Portuguese will stay beyond this season, with reports suggesting he could leave if the Red Devils miss out on a top four spot.
"Ronaldo looked angry today, he played with a lot of aggression," Keane said on Sky Sports.
"He's obviously upset with whatever happened last week [missing United's 4-1 loss to Manchester City], but he showed today what we have seen over the years.
"The guy's a genius, his three goals were fantastic. He brings so much to the team, why people think he cannot bring goals or value...
"For as great as he is, the next challenge for Ronaldo is to make this United team compete again for league titles.
"Ronaldo is a huge part of the rebuilding of the club. When he's producing like that, you need Ronaldo in your team.
"Ronaldo has been doing this throughout his career, he's been doing it for years. I think he can do it for another season."
