The Light Blues made a rapid start at Tannadice as Tom Lawrence broke the deadlock inside five minutes.
And the away side had further opportunities to stretch their advantage in a first half where they smothered Dundee United, but the second goal never arrived.
It set up a second half in which the hosts naturally gained belief, yet Clement’s side were able to keep their opponents at arm’s lengths and nervy moments were few and far between.
Speaking to RangersTV after the triumph, the Gers boss said: “It’s a well-deserved victory.
“It’s a hard place to come to, which everyone said before the game and it’s true.
“They’re coming out of a very good dynamic with gaining promotion and a really good start to the season.
“We started the game very well in the way we trained and we did really good things with and without the ball.
“In the second half, we could have used the moments better to score a second or third goal, but we didn’t give away anything. That was also important to keep this clean sheet.
“I’m very pleased if you see how many new faces are in the team and how many new ones came on in the second half; some of the boys don’t even know all the names of their teammates yet.
“It’s good that fans start to see we could bring in a few really young lads with a lot of potential for the future.
“It gives a lot of hope and belief in the next months and years to come.”
United threw several deliveries into the Rangers penalty area in the second half in an attempt to draw level.
However, the Gers stood up to the physical test and the most threatening moment from the hosts arrived from Kristijan Trapanovski who struck the post via a deflection off Connor Barron.
Clement continued: “It’s not only about nice football, but sometimes you need to dig in.
“We needed to win the fights and duels, which is what the team done. You need to do it at places like this, with the smaller pitches and defending throw-ins.
“The way we played quietened the supporters because we were in domination of the ball so they didn’t have a moment to get excited.
“We had around 80% of possession around 25 minutes and we had no chances against. It’s crucial to start with that concentration and intensity, with and without the ball.”
A return to Ibrox beckons for the Light Blues next weekend as they host the blue side of Dundee in the Premier Sports Cup quarter-final.
Clement says it’s a day that he’s been relishing for the last few weeks and has backed the Rangers support to bring the energy.
He added: “I’m so looking forward to that. It’s the feeling of coming home and it feels like ten years.
“If you see the energy of the fans today, it will be ten times more people. I cannot wait for that, and the players are the same.”