Wales international Amos makes his playing return at outside centre for the fixture at the Galway Sportsground, after recovering from the dislocated elbow he sustained at Leinster in mid-September.
Lock Davies comes into the second row to partner Lewis Evans, with Matthew Screech ruled out with an ankle injury.
The final change made by Head Coach Bernard Jackman is a first Guinness PRO14 start of the season for Wales Under-20 international Basham, who impressed off the bench in Belfast last week.
Experienced hooker Richard Hibbard will once again captain the team from the front row.
The replacements feature Dan Suter, with the prop pushing for a first league involvement this season. James Thomas is also called onto the bench.
“Hallam (Amos) is one of our most talented players and we want to get him on the ball as much as possible,” said Jackman, reflecting on the team selection.
“With the anticipated conditions in Galway, it will be hard to get into the wide channels and with Tyler Morgan away we can have a look at Hallam at 13 and get him involved.
“Taine (Basham) is a player we have huge hope for,” added Jackman. “We gave him league game time last season as an 18-year-old and we believe he has a very bright future.
“We have a lot of good backs rows at the region, but with Ross Moriarty and Aaron Wainwright away with Wales we can look at him.
“Huw Taylor had a slight ankle knock in the week, which prevented him training, and James Benjamin has a dead leg, which he suffered against Hong Kong. But Taine deserves this start and he certainly impressed us all off the bench in Belfast last week.”
Connacht come into the game on the back of a late loss to Ospreys last week and Jackman has warned his Dragons to prepare to face a backlash.
“They won’t have been happy with the previous game with Ospreys and they will want to put that right on home soil. We must expect a backlash,” he said.
“We have to play the conditions well. Jason Tovey starts at ten and we have to make sure we can manage the wind that is forecast because when Connacht get territory it is hard to keep them out.”
Dragons (v Connacht Rugby): Jordan Williams, Dafydd Howells, Hallam Amos, Jarryd Sage, Will Talbot-Davies, Jason Tovey, Rhodri Williams; Brok Harris, Richard Hibbard (c), Lloyd Fairbrother, Joe Davies, Lewis Evans, Harrison Keddie, Nic Cudd, Taine Basham
Replacements: Rhys Lawrence, Aaron Jarvis, Dan Suter, James Thomas, Huw Taylor, Tavis Knoyle, Jack Dixon, Adam Warren