Australia, chasing 160 in Bangalore, looked on course for victory as Ben McDermott hit a knock of 54 but the chase was derailed when the 28-year-old tried to hammer Arshdeep (2-40) for six and ended up holing out to long-off.
Once McDermott departed, Kumar (3-32) struck twice in quick succession to leave Australia reeling on 129-7 before Arshdeep bowled an excellent final over in which he gave away just three runs and picked up the wicket of visiting captain Matthew Wade.
After being put in to bat earlier, India needed a half-century from Iyer (53) to post a defensible total, with Axar Patel (31) also contributing runs. Australia's Jason Behrendorff and Ben Dwarshuis picked up two wickets apiece.
"I'm elated to be honest to see everyone chipping in and contributing for the team," Iyer said. "The shots, innovation has been on point and I could see Arshdeep's calmness when he was bowling the last over.
"It wasn't an easy wicket to bat on. I feel 160 was the right total, which we got eventually. To defend it was a brilliant performance from the team."
India, who won the series 4-1, next travel to South Africa for a multi-format series including three T20Is, three one-day internationals and two tests, while Australia head home for a three-test series against Pakistan.
