The 23-year-old has crossed the city divide on a Bosman free transfer after spending his entire senior career with the Dens Park club he grew up supporting.
However, Levein has described how Robertson's capture marked the end of a trail which began six years ago when the United boss was still with former club Hearts.
On two previous occasions, with Hearts and then Leicester, the ex-Scotland international defender offered money in a bid to land the talented youngster.
And he is thrilled to have finally captured Robertson for free.
He said: "The fact is I've been trying to sign Scott now since he was 17. I first saw him play for Dundee under-18s when I was at Hearts.
"I tried to sign him for Hearts and I tried to sign him again at Leicester and I offered money for him both times.
"Obviously I kept a very close eye on his career and how well he's been doing and I think he has improved, year-on-year.
"I'm finally delighted to have managed to get him to come and play for me and for Dundee United."