The three sports will now be put to the vote at the IOC session in Buenos Aires in September with only one of them winning a place on the 2020 Games programme.
The IOC is eager to revamp its sports programme in a bid to keep the Olympics relevant to a younger generation to attract viewers and sponsors.
Five other sports - karate, wakeboarding, sports climbing, wushu and roller sports - did not make the shortlist, meaning their chances of Olympic inclusion were dashed for at least another four years.
"It was never going to be an easy decision but I feel my colleagues on the (executive) board made a good decision in selecting baseball/softball, squash and wrestling to be put forward in Buenos Aires," IOC president Jacques Rogge said.
For IOC vice president Thomas Bach it was "a good mix between team sports and individual sports.
"Wrestling managed to convincingly present the changes they have undertaken, both on a sporting level as well as in the organisation," Bach, a candidate to succeed Rogge in September, said.
Wrestling was taken off the 2020 Games programme in February in a shock move but Wednesday's decision threw the sport an Olympic lifeline.
The sport, which traces its roots to the ancient Olympics in Greece and has featured in every modern Games since 1896 apart from the 1900 edition, has frantically scrambled to push through a string of reforms since February in a desperate bid to remain in contention.
The IOC has been heavily criticised for its decision to take wrestling off and should the sport now claim back its Olympic spot, a two-year process aimed at revamping the programme will have resulted in no change at all.
Wrestling comfortably made the cut on Wednesday after only one round, winning a majority eight out of 14 votes.
Baseball/softball, which received no votes in the first round, booked their spot next after beating karate nine votes to five in a head-to-head.
Squash clinched the final place on the short-list, grabbing eight votes compared to wushu's four and sport climbing's two.
Olympic inclusion guarantees sports millions of dollars of revenues and exclusion can stifle a sport's global growth.
The 26 Olympics sports at last year's London Games got anything from $14 million to about $47 million each, depending on their popularity, from the IOC alone as part of a revenue-sharing deal.
"All of those who helped us, I want to call on them to help us again because the match is not finished," said international wrestling federation (FILA) president Nenad Lalovic, who took over after the February exit.
"The second match starts now," said the Serbian, who jumped up and hugged associates in a packed conference room during the IOC announcement. "This is a small relief. The big relief will come in September. Everybody is fighting for his sport now."
Baseball and softball, which in 2005 became the first sports to be taken off the Games programme since polo in 1936, united into one federation and the move paid off as they grabbed a spot on the list.
"We got a chance to fight for our dreams," Don Porter, co-president of the World Baseball Softball Federation, told Reuters.
Porter was in charge of softball when they were thrown out of the Games. The sport was last played at the 2008 Beijing Olympics.
"We are in the seventh inning, we will now move to the ninth and I guess we need to hit a home run. Our young athletes had dreams and we can now help them get those dreams back," said Porter.
For squash, which battled to win a spot at the 2016 Olympics but lost out to rugby and golf, it was a long-awaited decision.
"This is a huge milestone in our quest to join the Olympic Programme and I would like to thank the executive board for the faith it has placed in squash," federation chief N Ramachandran said.
The IOC will elect the 2020 Games host city at its session on Sept. 7 and the new sport to be included in those Games a day later. Tokyo, Madrid and Istanbul are bidding to stage the 2020 Olympics.
- Sports & Recreation
- Buenos Aires