Abubakar's goal in the last moments of the second half came after Christian Ramirez gave Houston the lead in the first seconds of the half. It was also just the defender's third goal in four MLS seasons, and his first since joining Colorado during 2019.
The Rapids are still without a victory since the COVID-19 pandemic interrupted the season in March, but they have now drawn three consecutive games. Sam Vines picked up his first career MLS assist.
Ramirez's second goal of the season helped extend the Dynamo's MLS-best unbeaten run to six matches, even while their winning streak was snapped at three.
Houston was outshot 26-10 overall and 7-3 in efforts on target, yet looked on course for a win for much of the affair behind a strong night from goalkeeper Marko Maric, who made five saves.
But on a free kick from outside the penalty area in the dying moments, Vines chipped a weighted service beyond the back post toward the right flank of the penalty area.
Abubakar timed his run to beat Houston's high defensive line, took the ball down with his first touch, and drove a low finish past Maric from close range.
The goal avenged an error from the defender earlier, when he allowed Ramirez to slip out of sight just after the second half started as Memo Rodriguez controlled the ball on the right.
Rodriguez drove a long diagonal ball toward the back post, and Ramirez side-footed a first-time finish past William Yarbrough on Houston's first quality opportunity of the evening.
The Rapids dominated the opening stages, with Kei Kamara twice coming near an opening goal early in his first start since July.
In the eighth minute, he struck the upper right post with a curling effort from distance after carving out some space atop the 18-yard box. Ten minutes later, his effort from inside the right of the area forced Maric into a fine reflex stop.
Colorado continued to be the aggressor after Ramirez's goal, but seldom came as close to scoring until Abubakar's very late winner.