Orient had edged a goalless but entertaining opening 45 minutes with Moses Odubajo, Jimmy Smith and Sawyer all denied by good saves by the visitors' Ian MacLoughlin, who was the busier of the two goalkeepers.
However, it was Dons defender Shaun Williams who came closest to breaking the deadlock when his curled free-kick from 20 yards three minutes before the break fully extended Orient goalkeeper Jamie Jones, who tipped the ball round the post.
The O's moved in front after 81 minutes when full-back Gary Sawyer chose the moment to record his first goal for the club.
Martin Rowlands returned the ball into penalty box after a corner had been cleared and Sawyer collected the pass, turned and drilled a volley low into the corner of the net.
As the visitors pushed forward in search of an equaliser, Dean Cox and Kevin Lisbie exploited the gaps at the back, combining neatly for the latter to slip the ball past MacLoughlin and secure the points for Orient, who registered their first home victory over MK Dons.