"We've been here and we've had hard games before. You saw what happened to Celtic a couple of weeks back when they were 1-0 up and dominating the game as well (United won 2-1 with two late goals).
"So we were wary of that in the second half but obviously getting that second goal gave us that bit of breathing space.
"Within the game, I felt we were working really, really hard.
"We were two, three-nil up and we're still pressing the full-backs.
"Those lads, particularly in the wide areas, have been working really, really hard."
Winger Beasley, who recently expressed doubts about his Ibrox future, turned in a terrific display and capped it with a stunning goal on 26 minutes - the first time he has found the net since the 2008 Scottish Cup final.
"It was a fantastic goal and it came at a good time," enthused Miller.
"I felt we were on top for the majority of the first half so to get that goal was really important for us."
The win in the rearranged fixture was a third emphatic SPL victory in a row for Walter Smith's men following last month's surprise defeat at Aberdeen and puts them one point ahead of second-placed Celtic.
Dundee United and Hibs deserve a great deal of credit for the consistency that they have shown so far this season
"It's massive; you want to be at the top of the table," said Miller.
"You'd much rather be at the top of the table than chasing like we were for the majority of last season."
Manager Smith was also pleased with his side's approach at Tannadice, saying: "The whole team are probably working harder now than they have done at any stage of the season.
"They've been doing really well in terms of that, not just in overall play - which has been good in recent matches - but their determination to work.
"That's been a big factor."
Fourth-placed United now trail Rangers by seven points and manager Craig Levein apologised to home fans after a tame capitulation.
However, the reverse was just a second defeat of the season for the Terrors and Smith is not writing off the challenge to the Old Firm.
"Dundee United and Hibs deserve a great deal of credit for the consistency that they have shown so far this season," added the Ibrox boss.
"The games that they play against Rangers and Celtic are going to be important games, but if they continue their runs against all the other teams in the league it's going to be close."
This page is best viewed in an up-to-date web browser with style sheets (CSS) enabled. While you will be able to view the content of this page in your current browser, you will not be able to get the full visual experience. Please consider upgrading your browser software or enabling style sheets (CSS) if you are able to do so.