Ambrose received a straight red card for a challenge on Michal Kucharczyk just before half time in a game that also saw Legia miss two penalties.
Celtic – who gave a debut to new signing Jo Inge Berget – had got off to the perfect start, Callum McGregor cutting in from the right to fire in a deflected long-range shot to open the scoring on just eight minutes.
However, the home side were level just two minutes later when Miroslav Radovic drilled the ball through Fraser Forster’s legs after slack defending by Adam Matthews.
Kris Commons blazed over the bar as Celtic sought to regain the initiative but Legia took the lead on 36 as Radovic scored his second after scooping Ondrej Duda’s headed knock-down over the line.
Ambrose’s dismissal added to Celtic’s problems and although they made it through to half time without conceding again, Legia dominated the second half.
Charlie Mulgrew’s late tackle on Ondrej Duda just before the hour mark saw the first penalty of the night awarded – Ivica Vrdoljak missing the target.
But in the 84th minute Michal Zyro headed in the third before Mulgrew conceded another spot-kick with a trip on substitute Jakub Kosecki, Forster saving Vrdoljak’s spot-kick.
And Legia then added a late fourth in the final seconds when Kosecki latched on to a pass from Lukasz Broz, giving them a three goal advantage ahead of next week’s second leg at Murrayfield.