Mauricio Pochettino's side took the lead early on through young striker Harry Kane, who made space in the penalty area to receive the ball off Andros Townsend and rifle the ball into the corner of the net.
That lead was doubled just after 10 minutes when Roberto Soldado scored from close range at the second attempt, knocking the ball into an empty net when goalkeeper Craig Gordon parried his initial effort.
Celtic grew into the game as time went on, pulling a goal back just before halftime, as midfielder Dylan McGeouch ran at Kyle Naughton before hitting a shot on goal that deflected past Brad Friedel and into the net.
Moments later Naughton had redeemed himself. He exchanged passes with Townsend down the left side before teeing up Lewis Holtby for a comfortable finish inside the penalty area.
Soldado departed at the break with an injury but Lamela, on at the interval, enhanced Spurs' lead with a goal just after the hour mark. He hit a shot from 25-yards that deflected past Gordon in the Celtic goal.
Spurs wrapped up the victory with two goals in the latter stages. Christian Eriksen curled a free kick into the top-left corner before Emmanuel Adebayor dispatched a penalty with minutes to spare.