Dawkins opened the scoring in the 34th minute when he tapped home from close range after good work from Massimo Luongo, skipper on the day.
Orient levelled within three minutes and then went ahead 10 minutes into the second half.
Coulibaly finished well from a fine pass from Dawkins, striking high into the net, to make it 2-2 in the 74th minute.
But once again the goalscoring joy was short-lived as the O's fired home what turned out to be the winner just two minutes later.
Goalkeeper Oscar Jansson kept the team within touching distance with a number of fine saves at 3-2 before Coulibaly, Jake Nicholson and John Bostock all went close to rescuing a draw, Bostock lashing just wide with the last kick of the game.
"I thought we were excellent at times and it was a typically cavalier performance that we try to produce," reflected Tim, our Technical Co-Ordinator.
"It’s all about producing players and not winning football matches at this level.
“I think Massimo was back to his best, Simon Dawkins did well and Jake Nicholson at right-back was excellent.
"The Orient players had a lot of experience, so it was good for the lads to play against them. It was a decent exercise."
Simon Dawkins and Souleymane Coulibaly hit the target as we took the lead, fell behind 2-1 and then levelled at 2-2 against a more experienced Orient side before the visitors fought back again to secure their victory in the rain at Spurs Lodge.