Arsenal completed the signing of the Portugal Under-21 international from Benfica last week when the left-back arrived at the club for a fee believed to be worth £8m.
Starting his first game for the club, Tavares swept the hosts’ previous opener with a precise finish after 23 minutes.
Cedric Itten came on off the bench for Rangers in the second half and looked to have won his side with 15 minutes of a friendly on the left.
However, Eddie Nketiah saved the day for Mikel Arteta’s men with an equalizer in the 83rd minute.
It ensured the Gunners avoided a second straight defeat in Scotland following a 1-2 defeat to Hibernian earlier in the week.
Reacting to the new signing’s performance, Arteta said: ‘What a fantastic debut with his right foot!
‘He’s only been with us for a few days but you can already see his quality, his physicality and then the quality of the ball he has.
‘He’s getting along with the players very well, so it’s a good start.’
Speaking on Arsenal’s YouTube channel, Tavares is pleased with his impact and is enjoying the opportunity to perform in front of fans at the Emirates.
‘I feel good, it was a good game for me and for us. A good test,’ he said.
‘I think we played a good game and we learned a lot more.
‘I feel very happy to have my first gala with Arsenal. I’m so excited for this and today I did.
‘It’s good to play with the best, [Pierre-Emerick] Aubameyang and [Alexandre [Lacazette] are very good players.
‘I’m so excited to be playing in front of the fans at the Emirates, I can’t wait for this.’