Their second-round match was halted by rain on Wednesday with Fabbiano leading 7-6 (7), 6-3, 5-6.
Tomic said he was all set to get on a plane and head home just before learning of his late first-round inclusion and believes he can make an impression in his favourite event of the year.
She broke the teenaged world No. 109 in the opening game of the match and then again in the fifth game to storm through the first set.
The 17-time Grand Slam champion dug in though and managed to avoid being dragged further into the afternoon, claiming victory when Kukushkin's forehand smacked into the net tape.
Muguruza admitted it had not been her day.
Reacting to the call, Kyrgios said to the umpire: "What, after I hit it?" It affected my knee a little bit.
Earlier, the scoreboard said 6-4, 6-3, 6-4 to Rafael Nadal but it was an uneasy two-and-a-half hours for the Spanish world No. 1 as he laboured past Kazakhstan's Mikhail Kukushkin into the third round.
Djokovic, the three-time champion, needed treatment on his left thigh on Court Two but still had too many weapons for Argentina's Horacio Zeballos, winning 6-1, 6-2, 6-3.
Gavrilova dismissed American teenager Caroline Dolehide 6-2 6-3 to set up an all-Australian showdown on Thursday with Stosur, a 6-4 7-5 victor over China's Peng Shuai.
Cilic was two sets to the good when rain struck on Wednesday evening, but surrendered a vital service break in one of the last acts before play was postponed until morning.
But the Croatian collapsed once the match resumed on Court One, Thursday and Pella took full advantage to set up a third round clash with Mackenzie McDonald.
Shardul Thakur to replace injured Jasprit Bumrah for England ODIs
He was a part of the Chennai Super Kings in the IPL where he was their leading wicket-taker with 16 scalps from 13 matches. Shardul has so far played three ODIs in which he has claimed five wickets.
An inspired Pella capitalised on a flat performance from Cilic, beaten by Roger Federer in the final 12 months ago, raising his first-serve percentage and making just 18 unforced errors as the 2014 US Open champion toiled.
"He's good", Nadal said of de Minaur. Playing on grass, obviously having 10-plus aces in a match, it's not a common feeling that I get to experience a lot. I am very happy the way that I played today.
Nadal, the 2008 and 2010 champion, was tested by world number 77 Kukushkin, a player who is coached by his wife.
He was also hit with two time violations - one before the match had even started as he sat courtside.
I don't see a watch here. I don t see that countdown. "It's more hard for me to understand what's going on", said 11-time French Open champion Nadal, who is chasing an 18th major.
Juan Martin del Potro showed he is shaping up nicely for tougher matches ahead with a 6-4, 6-1, 6-2 romp past Spanish veteran Feliciano Lopez to reach the third round.
John Isner, the United States ninth seed, saved two match points, fired 64 aces and 88 winners to defeat Belgian qualifier Ruben Bemelmans 6-1, 6-4, 6-7 (6/8), 6-7 (3/7), 7-5 while Nick Kyrgios, seeded 15th, unleashed 42 winners to see off Robin Haase of the Netherlands 6-3, 6-4, 7-5.
Kyrgios received a code violation for swearing and also had to have the foot-fault rule explained to him by umpire James Keothavong in his 6-3 6-4 7-5 win over Robin Haase.
Japanese 24th seed Kei Nishikori overcame a first set defeat to beat Australian lucky loser Bernard Tomic 2-6, 6-3, 7-6 (7), 7-5. Japan's Naomi Osaka advanced to the third round by defeating Britain's Katie Boulter 6-3, 6-4.