Defending champion Cameron Smith was in danger of joining them but the Australian played a superb six iron to he final green and holed an eagle that sent him through to the weekend action.
"It was nice," Smith told reporters.
"It would have been very frustrating if I had missed out on the weekend. Defending is one thing, but just the golf that I feel like I'm playing and to miss out on the weekend would be very frustrating."
Smith is still 12 shots behind leader Brian Harman.
"It's a big ask to make up 12 shots, but I feel like a good weekend can get me right up at the top of the leaderboard," he said.
Mickelson, who lifted the Claret Jug at Muirfield in 2013, finished on nine over par and Morikawa, champion at Sandwich two years ago, was four over for the first two rounds.
Francesco Molinari, Shane Lowry, John Daly, Darren Clarke and Ernie Els were other former Open winners who missed the cut mark at three over along with major champions Dustin Johnson, Justin Thomas and Justin Rose.
Italian Molinari won the title at Carnoustie in 2018.
"It's complicated to have a very consistent career," he told reporters.
"I think everyone goes through ups and downs and you just do your best to navigate both the ups and the downs."
Fifty-year-old American Stewart Cink, Open champion in 2009, did make the weekend after adding a 73 to his opening 68 to finish at one under alongside 2016 winner Henrik Stenson of Sweden.
Two former St Andrews Open champions also advanced on two over par, South African Louis Oosthuizen, winner of the Claret Jug in 2010, and American Zach Johnson who lifted the trophy in 2015.
