18-year-old Kokkinakis battled past Lukas Rosol 4-6 2-6 7-5 7-5 6-3 before Tomic beat Jiri Vesely 6-4 6-3 7-6 (7-5).
Great Britain are in pole position to reach the Davis Cup quarter-finals following James Ward's epic five-set win over John Isner.
James Ward came through an epic to stun John Isner and give Great Britain a 2-0 lead in their Davis Cup first-round tie with the United States on Friday.
World number five Andy Murray gave Britain the advantage by dispatching Donald Young 6-1 6-1 4-6 6-2, but few would have expected Ward – who beat Sam Querrey in five sets at the same stage last year – to win against a player ranked 91 places above him.
And that possibility looked even slimmer when Isner surged into a two-set lead.
However, Ward stormed back and finally settled a match that lasted almost five hours, putting the hosts in pole position in Glasgow.
World number one Novak Djokovic's victory against Mate Delic was much more straightforward.
Djokovic put Serbia ahead in their tie with Croatia with a 6-3 6-2 6-4 success, with team-mate Viktor Troicki then producing a stirring comeback from two sets down to overcome Borna Coric 4-6 1-6 6-3 6-2 6-1.
Defending champions Switzerland, playing without Roger Federer and Stanislas Wawrinka who inspired them to the 2014 title, are level at 1-1 with Belgium following the first day in Liege.
Belgium, minus David Goffin due to a back injury, fell behind as Ruben Bemelmans lost 1-6 6-7 (6-8) 6-4 6-0 6-2, but Steve Darcis restored parity by beating Michael Lammer 6-3 6-1 6-3.
France took complete control against Germany in Frankfurt thanks to Gilles Simon and Gael Monfils.
Simon fought his way past Jan-Lennard Struff 7-6 (7-4) 2-6 6-7 (1-7) 6-2 10-8 before Monfils proved too good for Philipp Kohlschreiber in a 6-4 7-5 7-6 (7-4) win.
Australia lead Czech Republic 2-0 in their tie in Ostrava.
The 18-year-old Thanasi Kokkinakis battled past Lukas Rosol 4-6 2-6 7-5 7-5 6-3 before Bernard Tomic beat Jiri Vesely 6-4 6-3 7-6 (7-5).
Kazakhstan and Italy are locked at 1-1 after the opening day in Astana.
Mikhail Kukushkin crushed Simone Bolelli 7-6 (8-6) 6-1 6-2, but Andreas Seppi responded for Italy with a 6-3 6-3 6-7 (5-7) 6-2 win over Andrey Golubev.
