Following Monday's match, Williams told the BBC she was "more relieved" to have won, admitting that Rus, her opponent, had put up a good fight.
Williams was not the most mobile from the baseline but she was able to dominate from deep, her power enabling her to dictate most of the rallies - the deep forehand cross-court a particularly productive weapon. Now that Im not playing doubles in this event, I have the day off, I think that will help..
Back in 2011, there was the freaky situation of Williams hiding in her panic room when testers arrived at her LA mansion, dialling 911 as she believed them to be intruders.
Apart from adjusting to her role "as a working mom", the seven-time champion also had to get used to hearing something "new" as she was addressed as "Mrs Williams" for the first time at the grasscourt major following her marriage to Reddit co-founder Alexis Ohanian previous year.
"Looking back I'm glad I had that win".
Caroline Wozniacki needed less than an hour to claim her first win of the grand slam, but last year's semi-finalist Magdalena Rybarikova bowed out. "I'm going to do whatever I can to make sure that I'm prepared, in the best shape possible, hopefully playing some good tennis by the time the US Open comes around".
"I really don't like being away from her. I remember I struggled early on a lot", said Federer. "I just won that a couple of days ago", the Australian Open champion said. I don't go out there expecting to do well, see what happens.
Williams is bidding to equal Margaret Court's record of 24 Grand Slams at the end of the fortnight, nine months after giving birth to daughter Alexis Olympia Ohanian.
Vekic, who sealed the upset on her fourth match point, faces Slovakia's Viktoria Kuzmova or Sweden's Rebecca Peterson in the second round. "I didn't serve, actually, 'til I got here, to be honest".
Karolina Pliskova also had a battle on her hands to get past British hopeful Harriet Dart in three sets. "Until I read that article I didn't realize it was such a discrepancy with me as well as against the other players that they listed, at least the American players - both male and female".
Williams found herself in a bit of a jam against 105th-ranked Arantxa Rus of the Netherlands, down by a break in the second set on a windy afternoon. Something that I would normally be okay with, normally have, 'Oh, this is this.' It's just getting all back to the hang of things, getting used to different circumstances.
Former World No.1 Victoria Azarenka came through in straight sets, 7-6 (7-4) 6-3, against Ekaterina Alexandrova while 2015 quarter-finalist Madison Keys eased past Ajla Tomljanovic 6-4 6-2 in 63 minutes.
The Pole saved six match points to win 6-3 4-6 7-5 and scrape through to the second round.