The 26-year-old bagged a double in Glasgow's 60-3 win over Newport last weekend - the only one to do so with the six other tries coming from differing players.
That was the Warriors' fifth successive RaboDirect PRO12 win dating back to just before Christmas when they won the first of two clashes with Edinburgh to lift the 1872 Cup.
Glasgow welcome RaboDirect PRO12 leaders Ulster, who have lost just twice this season, next with seven points separating the Irish outfit in first and the Warriors in second.
Ulster ran out 18-10 winners on the opening evening of the season and Swinson admits he doesn't care who scores or who gets the attention as along as Glasgow succeed.
"All that matters to me is the team getting the result that we want. That will certainly be the case against Ulster," said Swinson.
"We are enjoying a good spell of form going into what will be a very difficult game and are confident we can get the win.
"Quite a lot went right against the Dragons. But we were especially good at the breakdown. We were a lot more physical in that key area than we had been.
"We were also clinical in our finishing. We ran hard in the backs. Those are two areas we will have to be good in against Ulster.
"We have played them three times this season and have lost three times. But the good thing is we know what mistakes we made against them before and where we need to improve.
"Ulster are one of the top teams in this league and in Europe. All aspects of our game have got to be good. But we will have to be aggressive at the breakdown and clinical in our finishing."