Robertson will now play the winner of the Mark Williams-Ronnie O'Sullivan match for a place in the semis.
The Australian got the better of a scrappy frame at the third time of asking but Gilbert hit back strongly with breaks of 69 and 121 to narrow the score to 11-8.
A break of 60 proved sufficient for Robertson to move within a frame of victory and he edged a nervy final frame with a 43 break to book his quarter-final berth.
Resuming 5-3 ahead overnight, Robertson ran in breaks of 48 and 34 to move 6-3 clear only for world number 68 Gilbert to recover to 6-5 behind with efforts of 45, 43 and 75.
Australia's Robertson assumed control from that point onwards with the 2010 champion contributing knocks of 76, 77, 82 and 92 to leave himself three frames short of the winning post and a match against fellow former world winners Williams or O'Sullivan in the last eight.
Gilbert won a scrappy final frame of the session, but the Tamworth player must win seven of a possible nine frames to extend his excellent run at the tournament when they play to a finish from 7pm.
"10-6 no complaints," said Robertson on Twitter. "Looking forward to tonight!".
Neil Robertson hit two centuries in constructing a 5-3 overnight lead over David Gilbert in the World Championship second round in Sheffield.
The 2010 world champion trailed 3-1 to big break-building qualifier Gilbert - whose best performance prior to this run was reaching the last 16 of the 2009 Welsh Open and last year's Australian Open last 16 - at the mid-session interval, but began to motor in winning four straight frames to illustrate why he is among the leading candidates to carry off the event.
Runs of 106 in the sixth frame and an imperious knock of 131 in the final frame of the afternoon left Robertson cutting a content figure as he departed the Crucible arena. It was little wonder with the man from Melbourne unearthing the type of snooker that saw him claim the Masters in January.
World number 68 Gilbert had to win four matches to reach the Crucible before upsetting the 11th seed Martin Gould 10-8 in the first round. He showed few signs of nerves in contributing to a tidy session, and did little wrong for most of the opening frames.
With Robertson boasting a pot success of 93 per cent compared to Gilbert's 87 per cent, it was a session of fine break-building. Gilbert trousered the opening frame with a 53 before Robertson restored parity with 76 in the second.
A 76 in the third frame after a re-rack saw Gilbert move 2-1 to the good before a superb cut on a red to a middle hole handed the Englishman the platform to lead by two frames heading for the tea break aided by a 44 break.
Robertson set out his own manifesto against the Tamworth player by swarming all over his opponent in the fifth and sixth frames as knocks of 84 and 106 saw him level at 3-3. Gilbert constructed a 45-0 lead in the seventh frame with Robertson responding with 57 before he missed the final yellow trying to develop the green off a side cushion.
Gilbert fluked the yellow trying to play safe, but left himself snookered. He failed to leave the green safe as Robertson cleared up to the black for a 4-3 lead.
With Robertson now in full flow, he applied the winning thrust with a closing run of 131 after Gilbert had left him an easy opening red.
It gives Robertson plenty of heart before they resume on Friday morning. The first man to 13 reaches the last eight with Robertson in line to face Mark Williams or Ronnie O'Sullivan in the quarter-finals.