The pair have both signed three-year contracts after completing medicals with the Scottish Championship club.
Tavernier and Waghorn will be reunited with former Latics defender Rob Kiernan, who moved to Ibrox earlier in the summer, along with former Hearts captain Danny Wilson and ex-Livingston winger Andy Halliday.
The 25-year-old Waghorn, who has had spells at Sunderland and Leicester, scored eight goals during 18 months at Wigan.
Tavernier spent the second half of last season on loan with Bristol City, who were promoted to the Championship during his time their.
The 23-year-old right-back started his career at Newcastle and has also had loan spells with Gateshead, Carlisle, MK Dons, Sheffield Wednesday, Shrewsbury and Rotherham.
Tavernier, who is likely to make his debut for the club in the Petrofac Cup against Championship rivals Hibernian, claims he had no doubt about moving to the second-tier of Scottish football.
“Rangers are a massive club and as soon as there was interest there was no thinking about it,” Tavernier said.
“It is a great opportunity to play up here and the recognition that you can get playing for Rangers is huge.
“You get different kinds of pressure with a promotion race, some lads feel it in different ways, and for me I like to play with a bit of pressure, it always makes me perform better and I enjoy it.”
Waghorn insists the influence of Mark Warburton and assistant David Weir was vital, after they took Brentford to fifth in the English Championship last season.
“I know the way they played with Brentford having played against them a few times,” Waghorn said.
“I hadn’t really met them on a personal level, but when I sat down with them and they got across their ideas and how they want to play, it was brilliant.
“It’s exactly how I want to play and how I see my football going. It was a no-brainer.
“I see myself being here for the long-term. The club has been through its troubles, but hopefully it is now on the right way. We have a good gaffer in and a good set of coaches.
“It’s a project, and with the players he has brought in there is only one way the club is going to go and that’s forward.
“So I’m really proud, really pleased, and grateful to be a part of it. Hopefully I can bring my best and do my best for the club.”