The free-scoring Dons captain took his season's tally to six goals, while substitute Jemal Johnson netted at the death to condemn Terry Butcher's Bees to their second successive defeat.
Milton Keynes created the better chances in an evenly contested start, and Andrews blazed Lloyd Dyer's lay-off over the bar from 16 yards.
Soon after Leon Knight intercepted Matt Heywood's poor back pass but was denied by a good block from Brentford keeper Ben Hamer.
As the Dons began to dominate, Andrews grabbed his fifth goal of the season to put them ahead in the 24th minute.
The visitors' skipper unleashed a thunderous low 35-yard drive that squirmed underneath the dive of the red-faced Hamer and into the net.
Milton Keynes controlled the remainder of the first half and Brentford's John Mousinho wasted the Bees' only real chance on the counter-attack when he burst onto a loose ball and fired over the bar from 16 yards.
Knight missed a great opportunity to double the Dons' lead within seconds of the re-start when he latched onto Aaron Wilbraham's defence-splitting lay-off and poked his effort into the side netting from 10 yards.
At the other end, Brentford striker Charlie Ide was denied by a brave close-range sprawling save from Dons keeper Willy Gueret as the hosts finally sprang into life.
But 11 minutes from the end Andrews made it 2-0 when he slid in to convert substitute Johnson's low cross from close range.
And Johnson sealed a comfortable win for the Dons five minutes into stoppage time when he rifled the ball past the rooted Hamer and into the corner of the net from 22 yards.