The skipper netted the second goal in the rout, with Gers setting a new Scottish top-flight record by opening a season with seven clean sheets.
Speaking to RangersTV, Tavernier explained: "I am absolutely delighted about. I think, first and foremost, to break that clean sheet record is a proud moment for the club and the players.
"To come into the league and not concede yet is a massive achievement. Our first and foremost objective before we went out was to keep a clean sheet, and while there were a couple of moments, we were just delighted to push it over the line and get that record.
"I thought we were great. Dundee United came after us and made it a little bit tricky at the start of the game.
"They came and pressed us, but I think we grew into the game and built through the press and created a lot of chances.
"I thought a lot of players could have had more goals but I am delighted that the subs who came on - Alfredo [Morelos], Scotty [Arfield] and Glen [Kamara] made a big impact, as did Calvin [Bassey].
"That is what is requested when the manager gives someone a chance to come into show it, I thought they were terrific.
"It has been a rollercoaster throughout, but this season we have started off really well and we just want to continue and help this team to be where we want it to be."
Rangers had to contend with four forced changes before and during the game as a result of injury – but the strength and depth of the team carried them through to victory.
Tavernier continued: "That comes from the top and the gaffer and the recruitment team that they have done an incredible job strengthening this team.
"At the start of the game, we lost Leon [Balogun], Fil [Helander] came into replace him and we brought George [Edmundson] onto the bench.
It’s on to the Europa League now for Gers, and a tie with Lincoln Red Imps on Thursday in Gibraltar.
Ahead of that, Tavernier added: "We can't underestimate the opponent and we have to give them respect. We know it is on astroturf and it will be hot, so we go over there with our game-plan and the right mindset to hopefully come away and be into the next round."