Raza hit 115 from 95 balls and was the penultimate wicket to fall after giving an under-strength Indian bowling attack a genuine scare. Zimbabwe were all out for 276 with three balls remaining, chasing India's total of 289-8 .
The visitors looked set for victory when they reduced the home side to 122-5 with all the regular scorers bar Raza back in the pavilion. But Raza shrugged off poor form in the first two matches to leave Zimbabwe needing 17 runs off 12 balls for victory. The tail could not deliver, and a gallant run chase by Zimbabwe came up short.
The win was set up by Shubman Gill who made his first international hundred. Gill smacked 130 off 97 balls, hitting 15 fours and a six. Ishan Kishan (50), Shikhar Dhawan (40) and Rahul (30) also contributed with the bat.
The breakthrough century for 22-year-old Gill came less than a month after he came agonisingly close, scoring an unbeaten 98 against the West Indies in Port of Spain.
Gill shared a 140-run third-wicket partnership with Kishan, who reached his half-century before being run out by Tony Munyonga. From the first ball of the final over Gill sought a boundary, but achieved height rather than distance off a Brad Evans delivery and was caught at long-off by Innocent Kaia.
Luck was twice on Gill's side as he survived a close leg-before appeal on 97 and was dropped by Victor Nyauchi on 110. Kishan struck six fours and Dhawan five fours while Rahul put behind him being out for one in the second ODI with a knock that included one six and one four.
Evans was the only bowler to impress, taking a career-best 5-54 in 10 overs.
Raza, regaining the form that brought him unbeaten centuries in two ODIs against Bangladesh, faced 95 balls and struck three sixes and nine fours. Among the Zimbabwe top order, Sean Williams (45) did best with an innings that included seven fours. Evans (28) shared a 104-run eighth-wicket stand with Raza.
Avesh Khan (3-66) took the most wickets among the Indian bowlers, but while Shardul Thakur (1-55) was expensive, he claimed the prized wicket of Raza. Spinners Kuldeep Yadav (2-38) and Axar Patel (2-30) were the best Indian bowlers on the day.