On Monday night Celtic had beaten Inverness 3-0 to put the pressure back on Aberdeen at the top of the table.
The young Dons took the lead after just ten minutes when Chris Maguire latched on to a Richard Foster cross to fire the ball past Ludovic Roy. Nine minutes before the break, Sandy Clark's side doubled their advantage when Andrew Considine headed an Ashley McInnes corner home.
Livi, who included ex-Don Noel Whelan and former Celt Colin Healy in their line-up, pulled one back five minutes into the second half when their third transfer window signing Steve Hislop lobbed the ball over Greg Kelly.
Aberdeen did not have too many chances in the second half, the best coming from a Maguire header that went wide, but they held out to secure the three points.