Ireland international Murphy will line up at full-back, with Josh Lewsey moving to the left wing in place of Shane Williams.
Cueto, who has scored eight tries in eight Tests for England, takes over from his Sale Sharks colleague Jason Robinson.
The other two changes are enforced, though, with Will Greenwood replacing centre Gavin Henson to partner skipper Gareth Thomas in midfield, and Stephen Jones taking over from Jonny Wilkinson.
Both Henson and Wilkinson are suffering so-called 'stinger' injuries to their neck and shoulder areas, joining a lengthy casualty list that also includes Charlie Hodgson, Ben Kay and Andrew Sheridan.
Lions head coach Sir Clive Woodward has retained the pack that produced an improved display in last Saturday's second Test.
Captain Thomas did not train on Thursday as he continues his recovery from a virus, but Woodward is confident he will be fully fit in time for the Test.
The pack is unchanged from Wellington, and six players - Lewsey, Thomas, Dwayne Peel, Gethin Jenkins, Julian White and Paul O'Connell - will have started all three Tests.
Defeat for the Lions at Eden Park would leave them suffering a first Test-series whitewash against any opponent for 22 years.
"Geordan Murphy has had a very good tour, and he was genuinely unlucky not to be included in one of the first two Tests, and I think the same applies to Mark Cueto, especially the way he played in midweek against Auckland," said Woodward.
"It was very tough on Shane Williams and Jason Robinson, but I think considering we were 2-0 down in the series, it was right to give a couple of players the chance who had been performing very well.
"It was also very tough on Shane Horgan, who I was thinking about starting with.
"Nothing against Jason Robinson or Shane Williams, I just believed these two players deserved a go. I am looking forward to seeing Cueto having a go, he is certainly one of the form European wingers, and he now gets his chance on the biggest stage.
"We've got one game to go. It is obviously a huge Test match for this side, and we would sincerely like to finish the tour with a win against New Zealand on Saturday.
"It was a case of getting everyone well-rested and just making the right calls after the Tuesday night game against Auckland," he added.
"The atmosphere in the camp is excellent. We are all very disappointed about the Test results, but it is just a massive opportunity for everyone involved. You don't get many chances to play New Zealand, and we've picked the strongest team we can put out."