Former champion Higgins, 41, had been 5-3 down after the first session but overturned the deficit to lead 9-7 on Saturday morning.
The first four frames of the evening session were shared but Scotsman Higgins clinched the next two.
Higgins will play Kyren Wilson in the quarter-finals.
Allen's best performance at the World Championships came in 2009 when he reached the semi-finals.
On that occasion, he was also beaten by Higgins who went on to beat Shaun Murphy in the final.
Northern Ireland's Mark Allen has been knocked out of the World Championships by John Higgins who won 13-9 in the last 16 at the Crucible.