Tavernier broke the deadlock with 32 minutes played, the captain scoring with a brilliant free-kick from a wide position which surprised Hibs goalkeeper David Marshall and beat him at his near post.
Hagi doubled his side’s lead with 55 minutes played, the Romanian firing home a shot with terrific accuracy from 16 yards after he saw an initial effort blocked.
Cantwell then added icing on the cake with four minutes remaining, the January signing finishing from close range after Marshall had used his legs to stop a deflected shot from Tavernier.
Hibs pulled a goal back in injury time through a Paul Hanlon header but it came too late for the home side who were second best for most of the match.
Rangers started positively and after winning an early corner kick, the ball dropped to John Souttar and his looping header clattered off the crossbar.
With 11 minutes on the clock a slick counter attack from the visitors led to Tavernier playing a pass into Fashion Sakala. The Zambian striker cut inside his marker with a clever piece of skill but his left-foot shot was straight at Marshall.
Hibs gradually started to get into the game, with a couple of tackles and slack passes from Rangers players getting the home supporters out of their seats.
On 24 minutes French winger Ellie Youan did well to create a yard of space on the left flank and his floated cross was met by Kevin Nisbet but his header was a foot wide of Robby McCrorie’s right-hand post.
The Light Blues, with Leon King and Hagi back in the starting eleven, put a lot into the first half hour but they were just missing that bit of quality required in the last third of the pitch.
On 32 minutes, however, James Tavernier produced a moment of sheer quality to put his side ahead and stun those in Easter Road wearing green and white.
After a foul on Todd Cantwell wide on the left wing, the Gers skipper stood over the free-kick with the players in the box, and everyone else in the ground, anticipating a cross.
Tavernier, however, had other ideas. The right-back whipped a superb shot towards goal and although Marshall got a hand to it he couldn’t stop it going in at his near post.
With four minutes of the first period remaining, Hibs came within inches of equalising. Nisbet did well to curl a low, teasing delivery to the back post but Josh Campbell, at full stretch, slid a shot past the post in front of the Rangers fans in the away end of the stadium.
Lee Johnson’s men were fired up for the start of the second period with Nisbet squirming a header past the post, with most fans in the stadium thinking it had gone in.
The game was halted briefly as Beale was forced to make his first change of the game eight minutes after the restart, with Scott Arfield replacing the injured Nico Raskin.
But any concern on the Gers bench soon turned to joy as, just two minutes later, Hagi made it 2-0. The Romanian saw his first shot blocked but when it came back to him he made no mistake, blasting a left-foot shot through a couple of defenders and into the bottom corner of Marshall’s net.
The 24-year-old enjoyed the moment as he celebrated his first goal of an injury-ravaged season, pulling on the badge on his jersey passionately in front of the travelling bears.
Despite being two goals up, Rangers kept pushing forward with intent, with Sakala firing a powerful drive at Marshall from a tight angle which he did well to hold.
The goalkeeper was again called into action in the latter stages, tipping the ball over his bar impressively after John Lundstram has tried his luck from distance.
Beale’s men got their deserved third goal with four minutes remaining when Tavernier’s deflected shot was stopped by Marshall but Cantwell was there to knock the ball home from close range.
Hibs got a goal back in injury time when Hanlon headed home but, in truth, Rangers dominated this contest and thoroughly deserved to be the team that came out on top, with youngster Arron Lyall handed a late debut.
HIBERNIAN: Marshall, Hanlon, Newell, Cadden (McKirdy 59), Jeggo (Doyle-Hayes 59), Nisbet, Stevenson, Youan (Hoppe 81), Fish, Egan-Riley (Miller 81), Campbell (Henderson 60)
SUBS: Johnson, Cabraja, Devlin, Rocky
GOALS: Hanlon 93
BOOKINGS: Doyle-Hayes
RANGERS: McCrorie, Tavernier, Yilmaz, Lundstram, Hagi (Lowry 74), Jack (Lyall 88), Cantwell (McCausland 88), Souttar, Sakala (Ure 88), King, Raskin (Arfield 53)
SUBS: McGregor, McLaughlin, Matondo, Devine
GOALS: Tavernier 32, Hagi 55, Cantwell 86
BOOKINGS: Cantwell