The League Two team had held their own for much of the first half but a goal in stoppage time from Chris Coyne put the Championship side ahead before second-half strikes from Warren Feeney and a Gary Alexander own goal saw them safely through.
But Orient refused to throw in the towel and Daryl McMahon grabbed a last-gasp consolation.
It was the home side who were quick out of the blocks as manager Martin Ling made three changes to the side who were beaten 4-1 at home by Rochdale on Saturday.
Teenage striker Efe Echanomi had the ball in the net after just two minutes but the effort was ruled out for offside.
Then Michael Simpson was denied by Luton keeper Marlon Beresford as Orient continued to press.
But Luton gradually took control and Enoch Showunmi wasted two glorious chances and Dean Morgan had a shot blocked before the Hatters finally broke the deadlock.
The goal came on the stroke of half-time when Orient failed to clear a corner and Coyne sent a clever lob from the edge of the box into the top corner.
After the break Morgan almost stretched Luton's advantage when he thudded a free kick against the bar, but the O's held out until the 71st minute when Feeney fired home a second.
Orient keeper Glyn Garner's poor clearance was intercepted by Showunmi and after he did the hard work Feeney smashed the ball into an empty net.
Orient's misery was complete five minutes later when skipper Alexander sent a bullet header into his own net when he tried to clear a free-kick from the impressive Kevin Nicholls.
But Orient's endeavour was rewarded in stoppage time when McMahon seized on a loose ball and fired a vicious shot from 18 yards past Beresford.