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.
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 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.
"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."