The Scotsman tied Robertson up well in snookers to take the opening frame with a finishing run over 40, but the 2010 world champion rallied with an early 54 break in the second before holding out for 1-1.
That would be as good as it got for the Australian, however, as a break of 90 and a late push in the tense fourth gave Burnett a two-frame lead heading into the mid-session interval.
And after Robertson's poor shot selection gave him the chance to snatch the fifth on the final colour trail, Burnett scored a 54 of his own en route to securing his place in the last 16.
Selby opens Wuxi title defence with win
Burnett, a finalist in China two years ago at the Shanghai Masters, will meet Mark Davis next in the second round after his win over local hero Ding Junhui later in the day.
Elsewhere in first-round action, Mark Allen recovered from dropping the opening set to Rouzi Maimaiti and powered to a 5-1 win.
Allen, fined earlier in the month for claiming that cheating in snooker was "a Chinese thing" after being dumped out of the World championship by Cao Yupeng, was quick and methodical in knocking out Chinaman Maimaiti and a last-16 encounter with Mark Williams or Tom Ford awaits.
Meanwhile Ken Doherty came back from 4-2 down to snatch a dramatic deciding frame over Lu Ning in a wild card qualifying match to earn a first-round tie with Shaun Murphy.
The Irishman, winner at the Crucible 15 years ago, took a 2-1 lead with a 102 century break only for Ning to respond immediately with a 101 to leave the two qualifiers level heading into the break.
And though things looked bleak for Doherty when the Chinese player claimed a second and third straight frame, "Crafty Ken" dodged the exit door twice to force a decider which came down to the final colours.
And just when it appeared Ning, who made the last 16 three months ago at the China Open, had snuck past the pink to cut an excellent brown on his way to sealing progress, it was ruled that he had fouled in the process and Doherty snatched the win from there.
Stuart Bingham and Peter Ebdon also went to a ninth frame in first round action before they were pulled from their table, after the first eight took a total of four and a half hours to complete.
Bingham would go on to defeat Ebdon in the postponed final frame to advance to the second round.
Bingham wins thriller in China
Allen fined £10,000 for 'cheat' slur