The scrum-half is one of six changes to the starting XV which lost the second Test 27-9 to the Springboks in Cape Town at the weekend.
Scotland wing Duhan van der Merwe retains his place on the left wing but fellow Scots Stuart Hogg and Chris Harris drop out of the squad altogether.
Finn Russell is named on the bench after recovering from an Achilles injury but Dan Biggar retains the No 10 jersey. The Lions looked desperately short of a creative spark in the two-try defeat and there had been calls for Russell to start.
With the series tied at 1-1, Lions head coach Warren Gatland has rung the changes.
The tour’s leading try-scorer, Josh Adams, makes his Lions Test debut on the right wing, replacing Anthony Watson.
Liam Williams replaces Hogg at full-back, while Bundee Aki is picked at inside centre, with Robbie Henshaw switching from 12 to 13 at the expense of Harris.
Price started the first Test, which the Lions won 22-17, and was unfortunate to be dropped to the bench for the second match. He returns to partner Biggar at half-back.
In the pack, there are two changes to the front-row with injury-free Wyn Jones taking his place on the loosehead to make his Lions Test debut alongside Ken Owens who is promoted from the bench to start his first Test of the Tour at hooker. Tadhg Furlong continues on the tighthead to win his sixth-successive Test cap.
Rory Sutherland, the Scotland loosehead who started the first Test and was on the bench for the second, drops out of the 23.
Alun Wyn Jones retains the captaincy to win his 12th consecutive cap and moves alongside Graham Price and Mike Gibson in the list of most Lions Test appearances.
Alongside Russell on the bench, there are call-ups for Adam Beard and Sam Simmonds who both feature for the first time in the Test Series.
Kyle Sinckler is included among the replacements but his participation is subject to the outcome of a disciplinary hearing after he was accused of biting in the second Test.
Scotland prop Zander Fagerson could come into the squad if Sinckler is suspended.
British & Irish Lions team (v South Africa, third Test, Saturday, Cape Town Stadium, 5pm BST)
15. Liam Williams (Scarlets, Wales)
14. Josh Adams (Cardiff Rugby, Wales)
13. Robbie Henshaw (Leinster Rugby, Ireland)
12. Bundee Aki (Connacht Rugby, Ireland)
11. Duhan van der Merwe (Worcester Warriors, Scotland)
10. Dan Biggar (Northampton Saints, Wales)
9. Ali Price (Glasgow Warriors, Scotland)
1. Wyn Jones (Scarlets, Wales)
2. Ken Owens (Scarlets, Wales)
3. Tadhg Furlong (Leinster Rugby, Ireland)
4. Maro Itoje (Saracens, England)
5. Alun Wyn Jones – captain (Ospreys, Wales)
6. Courtney Lawes (Northampton Saints, England)
7. Tom Curry (Sale Sharks, England)
8. Jack Conan (Leinster Rugby, Ireland)
16. Luke Cowan-Dickie (Exeter Chiefs, England)
17. Mako Vunipola (Saracens, England
18. *Kyle Sinckler (Bristol Bears, England) – subject to outcome of disciplinary hearing.
19. Adam Beard (Ospreys, Wales)
20. Sam Simmonds (Exeter Chiefs, England)
21. Conor Murray (Munster Rugby, Ireland)
22. Finn Russell (Racing 92, Scotland)
23. Elliot Daly (Saracens, England)