Despite the sides going into the break goalless, Orient made the breakthrough five minutes after the restart when Bonne stroked home from a Jake Caprice cross, before the forward sealed the victory four minutes from time.
Having suffered a heavy defeat at Bromley midweek, the O’s made two changes as wingers Jobi McAnuff and James Dayton returned to the side in place of Romuald Boco and Josh Koroma who dropped to the bench.
Defender Jamie Sendles-White was also named in the squad for the first time since arriving this summer, replacing fellow defender Michael Clark among the substitutes.
Following a subdued start to the afternoon in which neither side managed to force any early shots on goal, Macauley Bonne had the first effort just before the 20-minute mark as he fired over after collecting a pass from James Dayton.
David Mooney then reacted well to reach a ball from the right and hook it back across goal but it was just too far in front of Jobi McAnuff to reach.
Woking looked the most threating from set-pieces and it the hosts skipper Richard Orlu was presented with two opportunities in as many minutes as he first headed wide at the far post before seeing a shot blocked by George Elokobi.
With just over 10 minutes of the half remaining, Woking forward Jamie Philpot was allowed to cut inside from the right and unleashed a curling shot towards the far post which had Charlie Grainger at full stretch before dropping wide.
As the half drew to a close Dayton was presented with the O’s best opportunity of the half when fouled 25-yards from goal, but despite curling his free-kick over the Woking wall his effort fell the wrong side of the post.
One minute stoppage time was shown and with seconds remaining Mooney went down under a challenge inside the area but instead of pointing to the spot referee Christopher O’Donnell booked the O’s forward for simulation as the sides went into the interval level.
Both sides returned for the second half unchanged, and it was Orient who started the brighter as Dayton stung the hands of Card’s goalkeeper Nathan Baxter on the 47th minute with a powerful drive from inside the box.
Seconds later midfielder Alex Lawless drilled an effort over the bar after being teed up by Mooney, though the O’s managed to break the deadlock with their next opportunity through a well-worked move.
Having latched onto Bonne’s header, Mooney switched the play out to Dayton on the right who picked out Caprice on the overlap and the full back drilled a fantastic ball into the box for Bonne to slide in and convert his second goal in as many games.
The goal was no less than what Orient deserved following their bright start to the second half, and having taken the lead the defence found themselves having to deal with a number of high balls towards Woking target man Inih Effiong.
With the final 20 minutes approaching, Head Coach Steve Davis made his first change of the afternoon when bringing on Craig Clay in place of First Team Captain Charlie Lee who had been booked in the first half.
Chances continued to be a rarity throughout a well contested battle between the sides, but with 78 minutes on the clock Dayton looked to catch Baxter out with a speculative shot from distance which dropped wide of the post.
Not comfortable with only a one-goal advantage, Orient had a brilliant chance to add a second on the 82nd minute a McAnuff deflected into the path of Mooney only for the forward to lose his footing and fire at Baxter.
The much needed second goal was to come in the 86th minute however, and it was the O’s number 9 Bonne who was on target once again when he converted from a tight angle following a surging run down the left.
Woking continued to try and find a way back into the tie, but were unable to trouble Grainger in the O’s goal as Joe Ward came closest with a strike in stoppage time which was comfortably held.
The O’s next in action on Saturday, August 26 when we return to E10 to host Eastleigh at the Matchroom Stadium.
WOKING: Baxter, Ramsey, Staunton, Orlu (C), Ralph, Ferdinand (Carter 62), Isaac, Philpot, Bawling (Banton 72), Ward, Effiong
SUBS: Mason, Young, Wynter
BOOKED: Ralph (21), Orlu (87)
ORIENT: Grainger, Caprice, Coulson, Elokobi, Widdowson, Lawless, Lee (C) (Clay 69), Dayton (Boco 90+3), McAnuff (Koroma 90), Mooney, Bonne
SUBS: Sargeant, Sendles-White
GOALS: Bonne (50 & 86)
BOOKED: Lee (23), Mooney (45+1), McAnuff (54), Lawless (63), Clay (70)
REFEREE: Christopher O’Donnell
ATTENDANCE: 2,885 (1,078)