Neil put Rangers ahead with a clinical finish from the edge of the area in the 22nd minute at Ibrox.
With no Celtic fans in attendance at the order of the authorities amid a ticket allocation spat between the Glasgow clubs, the whole stadium exploded in celebration of the midfielder's first goal since his close-season move from Sunderland.
Rangers' new striker Kevin Kelsy also got his maiden goal for the club when he fired home to double the lead in the 59th minute.
Neil added Rangers' third with another fine finish in the 75th minute to settle a surprisingly one-sided quarter-final.
Rangers boss Derek McInnes was able to enjoy his first Old Firm derby success after arriving from Hearts to take charge in the summer.
McInnes' men have won their last five games in all competitions to erase the frustration of their slow start to the season.
It was a first domestic defeat of the season for Scottish champions Celtic, who rarely threatened the Rangers goal.
Martin O'Neill's team will not have long to wait for revenge as the two sides meet at Celtic in the Scottish Premiership next Sunday, with visiting fans again banned.
Joining Rangers in the semi-finals are Hearts, Hibernian and Aberdeen.
Hearts beat Stenhousemuir 3-0 on Friday, Hibernian defeated Ross County 3-0 and Aberdeen won 3-1 on penalties after a 1-1 draw with Kilmarnock on Saturday.