The Jordanians committed handball offences late in both halves at the Estadio Pedro Bidegain - the latter seeing a red card dished out to Anas Bani Yaseen - and the eighth-ranked Colombians made them pay via spot-kicks to James Rodriguez and Juan Cuadrado.
A raised arm from Rajaei Ayed Fadel in the 40th minute lead to Rodriguez's goal, before Yaseen grabbed a goal-bound ball with both hands to see red in the 83rd minute and Cuadrado converted.
Mario Yepes' sixth-minute header from a corner was off target, but almost fell to Eder Balanta at the back post, only for the shot to have too much power.
Al Dumeiri was fortunate not to concede a foul when he shoved Colombia's Camilo Zuniga out of the way on the left byline, but the referee waved play on - and the defender's cross in was flapped at by Colombia goalkeeper David Ospina, whose save fell to Ibrahim, but he could not convert from point-blank range.
Colombia failed to threaten from several set-pieces throughout the half, but got the chance to take the lead due to a brain-fade from Jordan's Fadel.
Zuniga almost doubled Colombia's lead on 54 minutes, with his swerving strike from just inside the area coming back off the crossbar.
Jordan almost had the ball in the back of the net too - not that it would have counted due to offside - as Al Saify's headed into the left post from an Al Dumeiri cross.
Colombia almost had the sealing goal in the 80th minute, when Victor Ibarbo's cross found Adrian Ramos at the far post, only for the latter's flicked attempt on goal to find the side netting.
But moments later Pekerman's side sealed the victory from the spot again, as Cuadrado - another second-half substitute - coolly finished.
