Samardzija, the third major player dealt by the Athletics during an offseason roster purge, likely will be placed in the White Sox rotation behind staff ace Chris Sale and in front of Jose Quintana. Chicago also received minor league pitcher Michael Ynoa in the deal.
"Jeff is one of the elite right-handed starting pitchers in baseball, and we believe his addition to our roster gives us the potential for a formidable starting rotation," White Sox general manager Rick Hahn said in a statement. "His competitiveness fits well in our clubhouse, on the mound and in our city."
Samardzija moved into the Chicago Cubs' rotation in 2012 and has gone 26-42 as a starter with a 3.86 ERA. He went a combined 7-13 last season with a 2.99 ERA with the Cubs and A's. He was 5-6 in 16 starts in the American League, posting a 3.14 ERA.
He is expected to make between $9 million and $10 million in 2015, his final year of arbitration eligibility. The former Notre Dame wide receiver will hit free agency after the 2015 season.
White Sox outfielder Adam Eaton took to Twitter to express his excitement over the team's acquisitions.
The 23-year-old Ynoa went 4-2 with a 5.52 ERA and 64 strikeouts in 31 relief appearances for Class A Stockton in 2014.
Semien appeared in 64 games with the White Sox last season, hitting .234 with 10 doubles, six home runs, 28 RBIs and 30 runs scored.
Bassitt made his debut for the White Sox last season, going 1-1 with a 3.94 ERA and 21 strikeouts over six appearances.
Robertson agreed to a four-year deal with the White Sox on Monday that is worth $46 million, a source confirmed to ESPN.com's Jerry Crasnick. The club's bullpen was among the worst in the majors last season, plagued by injuries and struggles.
Robertson, who went 4-5 with 39 saves and a 3.08 ERA after taking over the closer role in New York last year from Mariano Rivera, will be the new closer for the White Sox.
Doug Padilla of ESPNChicago.com and The Associated Press contributed to this report.