The fourth-seeded Canadian put in a near faultless display against his Serbian opponent, sending down 35 aces and never dropping a service game en route to a 6-7 (4/7) 7-6 (7/4) 7-6 (7/4) win in three hours and 14 minutes.
Raonic, ranked 31 in the world at the start of this week, looked in the mood to cause an upset against the world number nine from the outset, forcing four break-point chances when leading 2-1 only to see his opponent successfully fend them off.
Raonic dropped just four points on his service and managed 10 aces in an hour-long opening set, but ran out of steam in the tie-breaker as Tipsarevic claimed a vital mini-break to seize the early initiative in the match.
Tipsarevic looked like he might even close the match out in straight sets in the second when he forced two break points when 4-3 ahead, but Raonic recovered with a pair of explosive winners to keep it on serve before prevailing in the second tie-breaker of the match.
Tipsarevic came close to finally unlocking Raonic's thundering serve at the start of the deciding set but again Raonic did well to save the break-point chance, and when Tipsarevic did likewise to avoid falling 3-5 down, another tie-breaker loomed.
But on this occasion Raonic was taking no prisoners, racing into a 5-0 lead on his way to taking the tie-breaker and the match.