Both sides have played some impressive football in recent weeks but, like every other Scottish Premiership side, they had to battle all the way against lower league opposition in the second round of the competition.
Thistle saw off Cowdenbeath after extra-time while United were several minutes away from the same fate at Dumbarton when substitute Nadir Ciftci scored his second goal of the night to earn a 3-2 victory.
United have won the Scottish Cup and lost in the League Cup final since Dillon arrived in 2007, as well as playing in several more semi-finals.
And the defender knows that their runs to Hampden have not always been plain sailing.
The 30-year-old said: "Regardless of what I say, the lads should want to go and win a cup. There are only three trophies that you are competing for and the chances of winning the league are very, very slim. That's the reality of it.
"But if you look at the teams that have won cups in the last few years, it's possible to go and win one.
"You have to make sure you are desperate to go on and desperate to get there.
"A lot of it is about hard work and determination. It's great to play well but I think when we went on those runs, it hasn''t always been flair and great play that has got us there.
"Yeah, that has definitely helped, but sometimes you just need to grit your teeth and get through it. It doesn't always have to be a great win.
"The Dumbarton game was typical of that. There is no doubt that Nadir's quality and the free-flowing football in that small amount of time got the goals, but sometimes you just have to get on with it."