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Murray shouted "wake up!" during one of many frustrating moments on Rod Laver Arena but a dozy display proved sufficient as the Scot sealed a 7-5 7-6 (7/5) 6-2 victory.
He will now face Russian teenager Andrey Rublev, the world number 152 who had earlier registered a surprise win over Yen-Hsun Lu.
Murray, walking out on court at just after 3pm local time, was not addressed as 'Sir' in the announcer's introduction but that will have been of little concern to the top seed, who insisted in his post-match interview that "Andy is definitely fine". Instead, the Briton is likely keen to continue where he left off last season as he chases his first title in Melbourne, where he has reached the final five times but never been crowned champion.
The 29-year-old knows he will have to play far better than this if he is to lift the trophy a week on Sunday, after a scruffy performance that included 27 unforced errors, three conceded breaks of serve and a tendency to adopt the kind of careful, cautious tennis that will run him into trouble against more proficient opponents.
Marchenko, ranked 95th in the world and wearing his cap back to front, certainly offered a stern opening test, his exuberant shot-making and refusal to give in courting much support from an appreciative Australian crowd.
But baking hot conditions, with temperatures reaching 32 degrees Celsius, took their toll on the Ukrainian, along with Murray's superior experience under pressure, which ultimately proved decisive.
Marchenko's early break felt like little more than Murray blowing out the cobwebs, particularly when the Scot broke back before taking a 5-3 lead. His shots, however, were loose and his energy lacking, and Marchenko came again, deservedly breaking a second time after picking up a deft forehand volley.
Murray did just enough to edge the opening set, when Marchenko over-reached on a forehand, but seeds of confidence had been sewn in his opponent, who seized another break early in the second.
Frustrated, Murray bellowed to himself, and perhaps to his box, "Wake up!" before adding, "This is a grand slam guys, come on". Murray often needs a dose of agitation to jolt himself into action, however, and so it proved as he broke back for 4-4 before a brilliant top-spin lob, clipped on the run, put him 6-5 ahead.
Marchenko, who had missed a break point in the previous game, extended Murray to a tie-break but he wavered under pressure as three missed forehands helped Murray to a 4-1 lead.
There was no way back as Murray converted his third set point to move two sets clear and while Marchenko took a toilet break, perhaps to refocus, the rest of the contest was one-way traffic. Murray broke twice to storm into a 5-1 lead in the third set and - when a final Marchenko forehand drifted long - his progress was confirmed in two hours and 47 minutes.
Speaking in a news conference post-match, Murray told reporters: "I don't think it was the best match, to be honest.
"The conditions there were pretty different to what we've been practicing. Last week's been pretty cool. A lot of days, it's been overcast. The temperature of the court is much cooler. When it's like that, the ball is bouncing a bit lower, a bit easier to control the ball.
"I was a bit tentative because of that. Didn't serve that well either. So you end up having to work really hard on a lot of your service games when it's like that. It just was tough."
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Murray continued: "I didn't move that well. That's how it felt anyway. But sometimes that can also be down to the conditions, as well.
"The ball's flying through the air a little bit quicker, so the ball is coming onto you faster than what it was the last few days. Maybe wasn't reacting as quickly as I would have liked.
"But maybe also nerves there first round as well. It's maybe normal to feel a little bit slow on your feet or a bit heavy-legged in the first round."
Read the original article on Eurosport: Andy Murray battles past Illya Marchenko in searing heat