Jon Daly scored a hat-trick in just over 13 minutes either side of half-time to put United five points behind Hearts with two Clydesdale Bank Premier League matches left.
United bounced back from last week's 4-1 defeat to Celtic to give themselves a chance of overhauling Hearts, who visit Tannadice on the final day.
United need to beat Rangers at Ibrox on Tuesday but they did that last month and Houston was encouraged by his team's display.
"We got rid of some silly mistakes, defensively we were solid," said Houston. "In the middle of the park I thought Scott Robertson and Morgaro Gomis bossed it, and Jon Daly scored the perfect hat-trick with left foot, right foot and header.
"So we got four goals and a clean sheet against a good Motherwell, although not the strongest Motherwell side."
Source: PA
Source: PA