United took control just before half-time at Tannadice as they opened up a four-point gap on the Lanarkshire side in the race for third place with a 3-0 victory.
David Goodwillie claimed the final touch when Andy Webster's 38th-minute header found the net and Prince Buaben capitalised on a rare John Ruddy error to make it two before the break.
Danny Swanson broke away to add a third in the 72nd minute as United showed there was no hangover from their Scottish Cup victory over Rangers on Wednesday.
"I'm absolutely delighted with the application and effort again," Houston said. "Motherwell dominated the first 15 minutes and it wasn't until the sending-off that the game changed a bit. We were clinical, we scored a couple of goals with our first real chances.
"After half-time we had to make sure we were organised and played sensible 10-man football - don't need to press centre-backs and just drop in and deny space.
"I can't credit the players enough because we made five changes."
Houston admitted Willie Collum's decision to send off Dixon for a high tackle on Jim O'Brien, moments after the former United winger had caught the left-back, had galvanised his players.
"I don't know if they thought it was an injustice," the United boss said. "I think Dickers was slightly later as the two of them went to challenge for the ball.
"I could have brought Dickers' sock up because it's ripped right down the shin area.
"I'm not meaning Jim has gone over the ball because he is an honest guy and we had him here. But I just thought it was the coming together of two players and Dickers is the one that got a straight red card.
"However, after that I just felt there were another few tackles that went amiss and there wasn't a yellow card in the game, which surprised me a wee bit."
Motherwell's 12-match unbeaten run came to an end after they failed to exploit superior numbers for the fourth time this season.
Manager Craig Brown said: "At 11-a-side we were as good if not better. The 10 should be an advantage to the team with 11 but we went down to 10 men against Falkirk and dug in and you saw what Rangers did with nine men. I was very disappointed we didn't cash in."
Ruddy had kept nine clean sheets in his previous 12 SPL matches but gifted United their second when he failed to control a backpass.
"I couldn't believe that," Brown said. "The first goal, Andy Webster wasn't properly marked. The marking is usually very good at corners.
"The second one was a gift too but United, to their credit, took them.
"Big John was desperate to redeem himself but he didn't have an opportunity. He has been wonderful all season so he is exonerated for that one uncharacteristic error."