The Light Blues were frustrated in a goalless draw despite dominating the match from start to finish.
Tavernier told RangersTV: "We are disappointed as we have dropped two points today. We have dominated the game and dominated the ball and limited Livingston's chances.
"But I don't feel we were ruthless enough going forward. I felt Borna [Barisic]'s free-kick was one of the best chances we had, and that was a set-piece.
"But we have got to test their 'keeper. This was the first game of the season for him and we have to test him a lot more.
"We know next week against Kilmarnock is going to be the same so we have a week to prepare, to work on the training field and work hard this week as we know it is going to be a similar game.
"We have got to be more ruthless going forward.
"We always feel we are going to create chances and throughout the whole game, through the 90 minutes, we thought we were going to create chances, especially late-on.
"We kept driving and we kept getting the ball up the pitch to try to get in those areas where we could get the crosses in and to try and get combinations.
Tavernier was asked about keeping a fourth clean sheet of the season from four games, but he wasn’t overly interested in that, continuing: "The scoreline isn't good enough for us. We are Rangers and we want to come here and get the three points.
"It's another clean sheet, but we have to continue that as a team in general to limit these teams' chances and that gives the forward boys the opportunity to come up with the magic and produce the goods. "But we have got to keep working hard and we have got to get three points next weekend."