Taylor's fumble return, which came after he stripped Jake Plummer on a sack from behind, was the third Dolphins takeaway on Denver's last three drives.
But the big story on defense was a second-quarter goal-line stand that preserved the Dolphins' 6-0 lead.
The Dolphins stuffed Tatum Bell for no gain on first and second down from the 3-yard line before Taylor was flagged for offside, putting the ball at the 1.
But pressure by Taylor led to an incompletion by Plummer. After Denver decided to go for it on fourth down, Zach Thomas led the charge as the Dolphins dropped Bell for a 1-yard loss.
Denver did cut the lead to 6-3 at halftime, but the Dolphins took charge in the third quarter.
After their first drive of the second half ended with Frerotte getting stuffed on a quarterback sneak on fourth-and-inches, the Dolphins came back with a 61-yard touchdown drive that ended with a 2-yard touchdown pass from Frerotte to tight end Randy McMichael.
The Dolphins increased their lead to 20-3 on their next possession, with a 68-yard drive that ended with Sammy Morris' 9-yard touchdown run.
A Wes Welker fumble set up Denver's only touchdown in the fourth quarter, but the Dolphins responded when they took the ball over at their 40 with 9:19 left after the ensuing kickoff went out of bounds.
Instead of just trying to kill some clock, the Dolphins went up top right away with Frerotte hitting Booker, who then outraced the Denver secondary to the end zone.
Saban became the third consecutive Dolphins head coach to win his first season opener with the Dolphins, following Jimmy Johnson in 1996 and Dave Wannstedt in 2000.