David Mooney had headed the O’s into an early lead with a second minute header from Bonne’s cross, and the forward then went from provider to goalscorer with a low finish nine minutes into the second half.
Jack Sammoutis pulled a goal back for the hosts on the hour, and despite both sides creating several chances in the final stages Orient came away with the victory.
Having recorded victories at both Harlow Town and Burnham Ramblers, Head Coach Steve Davis named a strong side for the trip to the DGS Stadium with midfielder Charlie Lee captaining the starting XI which included new signings George Elokobi and Craig Clay for the first time.
Prior to kick-off, a minute’s applause was held in memory of O’s supporter Frankie Bish who sadly passed away earlier this year. Frankie was a good friend of the Greenwich Borough management team and former Orient duo Gary Alexander and John Mackie who built up a brilliant relationship with him during their time in E10.
From the first whistle Orient looked to get off to a bright start, and were able to do exactly that when making the breakthrough inside two minutes.
Striker Macauley Bonne brought the ball forward from the halfway line and expertly picked out David Mooney with a cross for the forward to stoop home and head in the opener.
Having fallen behind the hosts looked to respond immediately, but O’s goalkeeper Charlie Grainger pulled off a brilliant save to deny Travis Gregory from inside the area.
The opening few minutes were proving to be end-to-end, and it was almost 2-0 after five minutes as Trialist B saw his low strike crack back off the post.
Bonne’s movement was causing the Greenwich defence several problems, and the lively forward then came close to getting on the scoresheet on two occasions, first seeing a shot tipped over the bar before firing over from the edge of the box moments later.
Craig Clay was then next to come close for the O’s, with the midfielder almost marking his first appearance with a goal as his curling strike fell wide of the far post.
With 10 minutes of the half remaining, Grainger was once again called into action as he thwarted Charlie MacDonald with strong block after the former Orient forward had got in behind the defence.
Mooney’s early header proved to be the difference at the break, and following the interval the O’s made three changes as Dan Happe, Henry Ochieng and the returning Jobi McAnuff came on.
The early stages of the second half failed to match that of the first, but with 54 minutes on the clock Orient managed to double their advantage as Bonne pounced on a defensive error to stroke home his third of pre-season.
McAnuff then dragged a shot wide from the edge of the area, but with an hour played Greenwich managed to half the deficit as Jack Sammoutis combined well on the edge of the area and drilled a shot beyond the reach of Grainger.
Head Coach Steve Davis then introduced seven new faces for the final third of the match which included a first appearance for the latest arrival in E10 as Romuald Boco replaced Mooney.
Goalkeeper Sam Sargeant was another of the seven to be brought on, and the young shot-stopper produced a fine save to deny the hosts an equaliser before Greenwich substitute Jesse Darko fired into the side-netting soon after.
Both sides spurned chances as the match approached the closing stages, and with three minutes Boco had an opportunity to seal victory but saw his effort from inside the area impressively parried away in what proved to be the final chance of the afternoon.
Next up for the O’s is a trip to AFC Hornchuch on Tuesday, July 25 (kick-off 7.45pm).
ORIENT: Grainger (Sargeant 61), Caprice (McLean 61), Trialist A (Clark 34), Elokobi (Happe HT), Widdowson (Trialist C 61), Clay (Trialist D 61), Lee (C) (Ochieng HT), Trialist B (McAnuff HT), Alzate (Koroma 61), Mooney (Boco 61), Bonne (Abrahams 61)
GOALS: Mooney (2), Bonne (54)