The Whites lead 1-0 at the break after Dennis Adeniran’s header, before Cauley Woodrow’s second half double put the visitors in the box seat.
Following the introduction of Cox and McCallum just after the hour mark, the duo brought the O’s to life and combined twice to set up a nervy finale, only for Fulham to just hold on.
The O’s made five changes to the side which drew 1-1 with Cheltenham Town on Saturday with the likes of summer signings Callum Kennedy and Liam Kelly making their first starts for the club, while Aron Pollock, Nicky Hunt and Sam Sargeant also came into the side.
Fulham showed their strength in depth though, making 11 changes from Friday night’s win over Newcastle United.
Both sides settled well into the game, without Sargeant or his opposite number Jesse Joronen having much to do in the opening stages.
Fulham skipper Scott Parker came closest to breaking the deadlock in the early exchanges, as a Whites corner wasn’t fully cleared and the England international’s first time volley was well held by Sargeant.
That sparked the game into life and Orient went on the attack almost immediately as Gavin Massey jinked his way in between tackles before seeing a shot deflected behind for a corner.
A lovely ball through then found Harry Cornick, however the winger fluffed his kick and it trickled wide as the O’s began to get on top of the encounter.
Not for the first time in a O’s shirt, Ollie Palmer was putting in a shift and it was his perseverance on 23 minutes which created the next opening.
He won the ball on the edge of the box, before squaring for Massey. The ball got tangled up in his feet and Cornick tried to make the most of the opportunity but his effort was blocked.
The move was still alive though as it fell to Sandro Semedo and his shot took a deflection behind for a corner.
However, despite that spell of chances, the O’s were made to pay on the half hour mark when Vigen Christensen’s corner found the head of the onrushing Adeniran and he thundered the header past Sargeant.
Palmer continued his positive showing and worked his way in off the left just before the break and squared for a late arriving Liam Kelly, but his effort was saved by Joronen as the O’s went into the break a goal down.
No changes were made at the break but it was the Championship visitors who went on to effectively kill the game off with two goals in the space of three second half minutes.
The first was a sublime long-range effort from Cauley Woodrow with very little Sargeant could do, before the second came as Pollock was pushed to the ground out on the right by Woodrow, and with no foul given the striker was free to run into the box and curl an effort past Sargeant.
Shortly after the third, Dean Cox and Paul McCallum made a welcome return to the first-team fold for the first time since cruciate knee ligament injuries, replacing Semedo and Cornick.
And they both played a part in pulling a goal back for the O’s with 17 minutes remaining as Cox’s corner in from the right was met by the head of his ACL buddy McCallum and past Joronen as the hosts had some hope.
That goal certainly saw the momentum swing the O’s way and a Hunt ball forward was played into the path of Palmer by Massey and his shot was pushed behind for a corner.
The duo of McCallum and Cox were certainly having a positive impact once more and they gave Orient more than a glimmer of hope as the pair combined for number two with nine minutes to go.
This time Cox delivered from the right and it found the head of that McCallum ahead, and he wheeled away and the famous salute celebration was back out once more.
Orient continued to push forward with the hope of forcing extra time, but despite plenty of pressure they were unable to turn that into a late third.
While the game ended it defeat, the O’s showed plenty of fight and spirit despite finding themselves 3-0 down at one stage, and will be hoping to carry on where they left off when they host Newport County on Saturday.
ORIENT: Sargeant, Kennedy, Pollock, Erichot, Hunt, Weir (Atangana 73), Kelly, Semedo (Cox 62), Massey, Cornick (McCallum 62), Palmer
SUBS: Cisak, Clohessy, Moore, Gnanduillet
GOALS: McCallum (73, 81)
BOOKED: Kelly (28)
FULHAM: Joronen, Sessegnon, Ream, Stearman, Kavanagh, Edun, Parker, Adeniran (Tunnicliffe 64), Vigen Christensen, De la Torre (Cairney 88), Woodrow
SUBS: Button, Malone, Smith, Ayite, Kalas
GOALS: Adeniran (29), Woodrow (51, 54)
BOOKED: Adeniran (30)
REFEREE: Andy Davies
ATTENDANCE: 3,855 (1,193)