Russell, 20, who was playing on the left flank, headed United into a 53rd-minute lead after Garry Kenneth had met Paul Dixon's corner. And 21-year-old Goodwillie slotted home in the final seconds after being set up by 18-year-old substitute Stuart Armstrong.
Houston said: "Johnny Russell scoring was so well deserved because he put in a great shift. The boy has a great chance in the game. He is not even playing in his best position yet because he was a right good striker and a good goalscorer through all the youth teams."
The victory ended a run of five consecutive draws for United and took them level on points with Motherwell, with two games in hand, in mid-table in the Clydesdale Bank Premier League.
Houston added: "It was great to see a young lad like Stuart Armstrong come on and be positive, driving forward into the box, and one where I thought he was going to pull the trigger, he laid Goodie in, who showed good composure and a great finish.
"It wasn't Goodie's best night but I thought if you keep him on the park, he is liable to score a goal, and he did."
Motherwell manager Stuart McCall suffered the third departure of his short reign before the game when Marc Fitzpatrick left the club after rejecting a new contract.
McCall felt his side had their fair share of possession without threatening too often, although Dusan Pernis made a great late save from Chris Humphrey. But he was disappointed with the way his side lost the opening goal.
McCall said: "All managers will tell you they are hugely disappointed to lose a goal from a set-piece. Kenneth is a big lad but it's the second balls you have got to be alert to and we weren't.
"After that we huffed and puffed. The 'keeper made an outstanding save from Humphs with five minutes to go and we just got caught pushing forward at the end."