A 1-0 win over Inverness Caledonian Thistle 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.
He said: "I didn't think it was a great game but I'm delighted to get the goal and the win.
"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.
"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."
Source: PA
Source: PA