Francis will start at tighthead prop with Samson Lee moving to the bench. Aaron Jarvis has replaced Evans on the bench.
Jake Ball has also been named on the replacements bench for Bradley Davies, who has also returned home after he sustained a knee injury in Wellington.
Inside centre Jamie Roberts starts after he sustained a head clash with Malakai Fekitoa in the second test. He has gone through the necessary protocols to allow him to return to action after admitting much of the game was a "blur".
Gatland's settled side have put pressure on the All Blacks for sustained periods in the first two matches and face a New Zealand team that is more experimental with three uncapped players in their squad.
"We have played some good rugby in the first two tests and we want to finish the tour on Saturday with an 80-minute performance that we are happy with," Gatland said in a statement.
"We improved going into the second test and fought to the end and this weekend we have to have the belief and confidence that we can take the victory.
"Playing under the roof in Dunedin will be a great atmosphere and a quick track and we need to take our game to the All Blacks."
Wales: 15-Rhys Patchell, 14-Liam Williams, 13-Jonathan Davies, 12-Jamie Roberts, 11-Hallam Amos, 10-Dan Biggar, 9-Rhys Webb, 8-Taulupe Faletau, 7-Sam Warburton (captain), 6-Ross Moriarty, 5-Alun Wyn Jones, 4-Luke Charteris, 3-Tomas Francis, 2-Ken Owens, 1-Rob Evans
Replacements: 16-Scott Baldwin, 17-Aaron Jarvis, 18-Samson Lee, 19-Jake Ball, 20-Ellis Jenkins, 21-Gareth Davies, 22-Rhys Priestland, 23-Scott Williams
(Reporting by Greg Stutchbury in Wellington; Editing by Toby Davis)
