A 1-0 win over Inverness gave the Terrors their fifth win on the bounce and guaranteed their place in the top half of the league after the split. Swanson, who scored the only goal of the match two minutes before the break, sees no reason why the Tannadice side cannot challenge the Jambos to finish best of the rest after the Old Firm.
"I didn't think it was a great game but I'm delighted to get the goal and the win," he said. "The aim at the start of the season was to get into the top six and we will keep fighting with Hearts for third now."
Swanson added: "We're not going to give up on them. We just need to keep winning games as you never know what can happen.
"We've had five wins in a row and we have got the players here to keep that run going."
Next up for Peter Houston's men is a Scottish Cup quarter-final replay against Motherwell on March 30.
And the 24-year-old believes United could still enjoy as successful a season as they had last term when they finished third and lifted the Scottish Cup.
He said: "It just shows you the strength of the squad. We have had a lot of injuries this season but we are still up there. We are still in the Scottish Cup and we will be looking to win that again."
The United players will now enjoy a few days' rest following a punishing schedule of five games in the space of 12 days.
Swanson added: "I haven't been playing in that many games so I just want to keep going but I'm sure the boys are happy to take a break. I think they deserve their break as there are a lot of them that have been playing every game."
Source: PA
Source: PA