The Lions led early on thanks to a deft lob from their forward Joel Nouble and the same player had another chance, as did Ayo Obileye, to add to their tally.
Rangers found the going tough against the West Lothian outfit and took time to settle, not testing Livingston’s debutant goalkeeper Shamal George often enough.
But Rangers substitute Scott Arfield levelled with a glancing header in the 72nd minute before skipper Tavernier quickly curled in a free-kick to the relief of the Light Blues support whose nerves were frayed.
Rangers will travel to Belgium for the first leg of their Champions League third qualifying round clash with Union Saint Gilloise on Tuesday with more optimism than they might have had they not grabbed those two crucial goals in two minutes.
Andrew Smith took in the match and gives the Rangers players a mark out of ten – with one new boy suffering a torrid afternoon.
1. Jon McLaughlin
On a day that seemed to provide indication he will be Rangers no.1, a header from Obileye in the first period was about all the Rangers keeper had to deal with. Cut an assured figure. 7 Photo: Alan Harvey - SNS Group
2. James Tavernier
The Rangers captain had far from his best day, but he delivered the match-winning moment, as is his wont, with a curling free-kick that wasn’t his finest, but sufficiently good to beat George. 7 Photo: Alan Harvey - SNS Group
3. Connor Goldson
Had a really awkward afternoon in dealing with Nouble but as the confrontation wore on he found a better rhythm. 6 Photo: Rob Casey - SNS Group
4. John Souttar
An inauspicious first league start for Rangers he was sleepy in allowing Nouble to drift off him at opener and didn’t ever look set. No surprise he departed as Van Bronckhorst rung changes to retrieve the situation with 67 minutes played. 3 Photo: Rob Casey - SNS Group