First-half goals from Paul Dixon and Lauri Dalla Valle gave United a comfortable advantage but Andy Kirk netted early in the second half to pull the home side back into it.
However, late goals from Johnny Russell and Mackay-Steven secured victory for the visitors.
It was understood before the match that Dundee United chairman Stephen Thompson had cut short a family holiday to hold talks with Houston after the match.
The visitors enjoyed much of the early possession and created the first chance of the game when some neat build-up play led to Ryan Dow striking wide.
Two minutes later Dunfermline goalkeeper Paul Gallagher was forced into an excellent low stop from Gavin Gunning's ferocious free-kick.
United grabbed the goal their play merited after 13 minutes when Willo Flood's free-kick was only partially cleared allowing Dixon to unleash a fantastic left-footed drive into the bottom right corner.
United doubled their lead five minutes later when Dixon's tame effort was deflected into the path of Dalla Valle who coolly slotted the ball beyond Gallagher.
The home side created their first chance of the game after 22 minutes but David Graham's powerful shot was off target and their miserable start to the game continued when midfielder Gary Mason was shown a straight red card following a reckless challenge on Russell.
The home fans were furious at referee Steve O'Reilly's decision but Mason appeared to have his studs raised in the tackle.
United were well in command and continued to search for goals. Scott Robertson passed up a good opportunity after 35 minutes when he shot over from Dow's intelligent cut-back.
It was the visitors who made the brighter start to the second half and in the opening minute Gallagher did well to stop Dow's shot from distance.
Dunfermline came close to pulling a goal back after 50 minutes when Kirk's cross clipped the top of the crossbar, giving the home fans cause for optimism.
United manager Houston made his first change of the game after 53 minutes when he replaced goalscorer Dalla Valle with Mackay-Steven.
The home side grabbed an unlikely goal 10 minutes into the second half when Paddy Boyle won the ball in the midfield and set up Kirk who slotted the ball beyond Dusan Pernis.
From there they began to grow in stature and Jim McIntyre introduced striker Steven McDougall in place of Boyle.
But it was United who continued to create the better chances and they came close to grabbing a third goal when substitute Scott Allen's excellent run into the box forced Gallagher into an excellent stop.
John Rankin almost turned the rebound in but Alex Keddie was on hand to bundle the ball out for a corner.
With 16 minutes remaining United notched a crucial third goal when Russell took full advantage of poor defending from Mackay-Smith's corner to slam home from close range.
Russell had a glorious opportunity to add to his tally minutes later when Dixon slid the ball across the penalty area but the midfielder struck the bar with his first effort and blazed the rebound over.
With eight minutes remaining United secured victory with a fantastic piece of breakaway football.
From a Dunfermline corner Flood released Russell and the midfielder's jinxing run ended with a cool pass to Mackay-Smith who tapped home a fourth.