Playing what he described as some of his best tennis "for a year and half" world No.3 Djokovic brushed aside Lleyton Hewitt 6-2, 6-4 and Ivanovic eased to a 6-4, 6-0 win over Alicia Molik. They joined forces to win the mixed doubles 6-7, 7-5, 10-6.
Serbia now need to win just one rubber on Thursday when they play Belgium, who whitewashed Kazakhstan 3-0 in the day's other Group A match.
Djokovic, who has been training with Britain's Andy Murray throughout the week as the build-up to this month's Australian Open intensifies, looked lean and mean as he bullied Hewitt from the baseline with some outstanding hitting.
Hewitt's only moment of hope came when he recovered a break to level at 2-2 in the second set but Djokovic broke in the 10th game to clinch an impressive win against the former world No.1.
Ivanovic survived an injury scare as she came through a tough first set before crushing Molik in the second.
She said she felt a sharp pain in her left knee in the third game of the second set but, though she planned to check it with the medical staff, she believed it would not be a problem.
Justine Henin, another former world No.1, continued her comeback from a six-month injury absence with a 6-4, 6-3 victory against Kazakh stand-in Sesil Karatantcheva.
Ruben Bemelmans then upset the much higher-ranked Andrey Golubev 6-4, 6-4 and the Belgians won the mixed 4-6, 6-2, 10-8.
In just her second match since Wimbledon because of a serious elbow injury, Henin was a little rusty but had too much experience for world No. 140 Karatantcheva.
Having arrived in Perth late on Monday as a replacement for the injured Yaroslava Shvedova, Karatantcheva had Henin worried when she broke to lead 3-1 in the second set before the Belgian reeled off five games.
In Brisbane, the women's draw at the Brisbane International was thrown wide open on Tuesday after top seeds Samantha Stosur and Shahar Peer were bundled out in the second round by unheralded opponents.
World No.6 Stosur was beaten by fellow Australian Jarmila Groth 6-2, 6-4 in the tournament's biggest shock, while earlier in the day the Czech Republic's Lucie Safarova outlasted second seed Peer 3-6, 6-1, 7-5.
With third seed Nadia Petrova losing on Monday, Frenchwoman Marion Bartoli is now the highest ranked player left in the women's draw.
Earlier, Safarova drew on boyfriend Tomas Berdych's Wimbledon heroics to deny Peer a place in the quarterfinals.
Safarova was alongside Berdych when he reached the final of last year's Wimbledon and said later that his efforts had inspired her to improve her own results.
Safarova saved a match point in the third set before eventually claiming a tense 3-6, 6-1, 7-5 win over Peer in a two-and-a-half-hour match that never reached any great heights.
Safarova will play fifth seed Anastasia Pavlyuchenkova in the quarterfinals following the Russian's 6-2, 7-5 win over Serbia's Bojana Jovanovski 6-2, 7-5.
Italy's Roberta Vinci earlier completed the women's first round when she beat American Christina McHale in straight sets 6-2, 7-6 (7/2).
In Doha, Roger Federer added to his repertoire of through-the-legs winners Tuesday on his way to beating Dutch qualifier Thomas Schoorel 7-6 (3), 6-3 at the Qatar Open in his first official match of the season.
After a sluggish first set, the No. 2-ranked Swiss hit form in the second. Leading 5-2 and faced with a ball, which changed direction after clipping the top of the net, Federer flicked it through his legs for a clean winner into the corner.
In the second round, Federer plays Swiss compatriot Marco Chiudinelli, who beat Reda El Amrani 3-6, 7-6 (3), 6-3.
Top-ranked Rafael Nadal also advanced, beating Karol Beck 6-3, 6-0.
"It was not an easy match," the Spaniard said. "I started well and then kept playing well. I think I gave a good performance." Nadal led 3-0 in the first set, and then broke Beck all three times in the second.
Nadal, who is seeking his first trophy in Doha, will face either Lukas Lacko of Slovakia and Pere Riba of Spain in the next round.
"I hope to do well here," said Federer, who won the tournament in 2005 and 2006. "I know (Rafael) Nadal is here and that makes this event very interesting." Also, fifth-seeded Ernests Gulbis of Latvia beat Victor Hanescu of Romania 6-3, 7-6 (6) and eighth-seeded Philipp Kohlschreiber of Germany defeated Andreas Seppi of Italy 6-2, 6-4.
In Chennai, top seed Tomas Berdych breezed into the second round of the Chennai Open on Tuesday with a 6-0, 6-1 victory over Portuguese opponent Frederico Gil.
The world number six took just 45 minutes to see off Gil, ranked 95 places below him.
"It was one of those matches. It does not happen always," said the Czech, who now takes on Croatia's Ivan Dodig.
Fourth seeded Frenchman Richard Gasquet also had little difficulty in getting the better of compatriot Edouard Roger-Vasselin 6-1 6-3, while Belgian seventh seed Xavier Malisse tamed Turkey's Marsel Ilhan 6-3, 6-1.
Alexandre Kudryavtsev dashed local hopes, beating Indian wild card Yuki Bhambri 6-2, 6-1, but the Russian qualifier faced a tougher task in the second round where Serbian sixth seed Janko Tipsarevic awaits.
