Becker saved all three break points that he faced in beating Slovakia’s Lukas Lacko 6-3, 6-2 and advanced to a second-round clash with defending champion and top seed Kei Nishikori of Japan.
Nishikori, who like the other top-four seeds received a first-round bye, is defending his Memphis crown after helping Japan beat Canada in the first round of the Davis Cup earlier this month.
Baghdatis, who reached the Australian Open final in 2006, beat US qualifier Rajeev Ram 6-3, 6-4 to book a date with Australian third seed Lleyton Hewitt.
Baghdatis, ranked 134th in the world, has not won two matches in a row since last year’s US Open, when he lost in the third round to Switzerland’s Stanislas Wawrinka, who went on to win the Australian Open last month.
Since then, Baghdatis has lost eight of his past 10 matches, but against Ram he was lethal, firing 10 aces and dispatching his opponent in just 75 minutes.
Karlovic blasted 14 aces to beat Aussie Matthew Ebden 7-5, 6-4 and advance to a second-round meeting with Spanish second seed Feliciano Lopez.
Meanwhile, Kazakh eighth seed Mikhail Kukushkin battled past Russian Teymuraz Gabashvili 7-6 (11/9), 7-5.
Kukushkin could face fourth-seed Lu Yen-hsun of Taiwan in the quarter-finals.
US sixth seed Sam Querrey squandered 18 aces and a match point in the decisive tie-breaker in losing to Russian Alex Bogomolov, whose 2-6, 6-4, 7-6 (9/7) triumph after two hours and two minutes gave him to a second-round match with American Ryan Harrison.
The winner of that tiecould face Nishikori in the quarter-finals.