The world No. 1 Djokovic disposed of Slovenian Aljaz Bedene 6-3, 6-2, 6-4, while the fourth-seeded Wawrinka moved on in similar fashion, topping Turk Marsel Ilhan 6-1, 6-4, 6-2.
Djokovic appeared to be tired and still recovering from illness after his match, saying he'd had a "rough two weeks health-wise but I'm getting up there."
His mood changed suddenly when a court-side interview turned to reflections of 2014, when he won Wimbledon for the second time, married his long-time girlfriend, Jelena, and the couple had a son, Stefan.
"Well those two events are the two most beautiful events I experienced in my life," he said. "Stefan, he's a blessing we received ... now I have even more motivation."
Djokovic titled here in 2008, then won three straight from 2011-13 before Wawrinka took the crown last year, stunning Rafael Nadal in the final.
"It's great, bringing me a lot of memories from last year," Wawrinka said of his return. "It was great to come back here feeling happy, happy with my game."
Up next for Djokovic will be Russian Andrey Kuznetsov, while Wawrinka will meet Romanian Marius Copil.
Fifth-seeded U.S. Open runner-up Kei Nishikori beat Nicolas Almagro 6-4, 7-6 (7-1), 6-2, eighth-seeded Milos Raonic was a 7-6 (7-3), 7-6 (7-3), 6-3 winner over Illya Marchenko and No. 9 David Ferrer overcame Thomaz Bellucci 6-7 (7-2), 6-2, 6-0, 6-3 at Melbourne Park.
No. 12 Feliciano Lopez trailed two-sets-to-one before rallying past American Denis Kudla 3-6, 6-2, 4-6, 6-2, 7-6 (7-5), while 13th-seeded Roberto Bautista Agut got past promising Austrian Dominic Thiem 4-6, 6-2, 6-3, 7-6 (7-5).
No. 16 Fabio Fognini was upset by Colombian Alejandro Gonzalez 4-6, 6-2, 6-3, 6-4.
Meanwhile, 17th-seeded Gael Monfils overcame fellow Frenchman Lucas Pouille 6-7 (3-7), 3-6, 6-4, 6-1, 6-4, 18th-seeded Frenchman Gilles Simon handled Dutchman Robin Haase 6-1, 6-3, 6-4 and 19th-seeded American John Isner dismissed Taiwan's Jimmy Wang 7-6 (7-5), 6-4, 6-4.
In other action involving seeds, Italian Paolo Lorenzi upended No. 21 Alexandr Dolgopolov 6-4, 6-3, 6-2; German Benjamin Becker bounced No. 25 Frenchman Julien Benneteau 7-5, 5-7, 6-2, 6-4; German Matthias Bachinger vanquished No. 27 Pablo Cuevas of Uruguay 7-6 (7-1), 6-3, 6-1; No. 30 Colombian Santiago Giraldo subdued Czech Jan Hernych 6-3, 6-2, 6-2; and No. 31 Spaniard Fernando Verdasco came back to best Brit James Ward 2-6, 6-0, 7-6 (8-6), 6-3.
Aussie favorite and former Melbourne runner-up Lleyton Hewitt moved on by beating Chinese Ze Zhang 6-3, 1-6, 6-0, 6-4.
Several other men reached the second round, including tall Pole Jerzy Janowicz, Finn Jarkko Nieminen, Americans Donald Young and Steve Johnson, and Canadian Vasek Pospisil, who outlasted American Sam Querrey in five sets, 6-3, 6-7 (5-7), 2-6, 6-4, 6-4. Young should have his hands full with the aforementioned Raonic here on Thursday.
Several stars will see second-round action on Wednesday, including four-time Aussie champion and former runner-up Roger Federer, the former titlist and two-time runner-up Nadal and three-time runner-up Andy Murray. The second- seeded former world No. 1 Federer will take on Italian Simone Bolelli, while the third-seeded former top-ranked great Nadal will meet American Tim Smyczek and the sixth-seeded Murray will tangle with Aussie Marinko Matosevic.
Also on the Day 3 schedule are seventh-seeded former Wimbledon finalist Tomas Berdych and 10th-seeded Grigor Dimitrov.