Cook Island international Taia, 32, has previous Super League experience with Catalans Dragons.
Saints Head Coach Kieron Cunningham said Australia-born Taia’s “big game experience” would help the squad.
“He is powerful and possesses a great offload,” Cunningham added.
St Helens have only played one game this season – a 6-4 victory over Leeds Rhinos – on the opening night of the 2017 Super League season.
Taia was named in the Super League Dream Team of 2015 whilst playing with Catalans, before returning to Australia where he began his career.
He captained the Cook Islands in the 2013 Rugby League World Cup and also appeared at Test level for New Zealand.
Cunningham said Greenwood’s move to the Gold Coast Titans was a “good opportunity” for the 23-year-old.
“We wouldn’t want to stand in the way of any player wishing to ply their trade elsewhere.
It’s always difficult to lose a homegrown player and personally I will be sad to see him leave. I’ve worked hard with Joe to develop his game and help him grow as a person - and he has done just that.”
Greenwood played 77 times for the Saints after joining from Saddlworth in 2012.
“Hopefully, one day after a successful NRL career, he will pull on the Red V again,” Cunningham added.