The Ghana-born Norway midfielder, signed from Rennes, marked his debut with a 22-yard effort midway through the first half, which proved enough to see off Dean Saunders’s League One side.
Although there was a start for captain Grant Holt, who has yet to score this season, Canaries boss Chris Hughton made the expected sweeping changes, with former Barnsley midfielder Jacob Butterfield starting.
Norwich, who have yet to win in the Premier League under Hughton, started brightly. Tettey showed his intent early on when he let fly from 25 yards, but his shot was just over.
Doncaster had a chance on 12 minutes after a quick break down the right when Martin Woods’s shot was blocked and Robbie Blake stabbed the rebound goalward, Bunn tipping it behind at full stretch.
Goals have proved a problem so far for the Canaries, and the hosts were out of luck again as first Harry Kane saw his effort blocked after a punch from Doncaster goalkeeper Gary Woods dropped to him in the six-yard box, before full-back Marc Tierney had a 20-yard strike tipped over.
Rovers soaked up plenty of pressure, but were dangerous on the break and midfielder David Cotterill flashed a low shot across goal.
Norwich took the lead on 26 minutes, when Tettey found his range, lashing a fine curling effort from 22 yards in off the far post. And the Canaries continued to press.
Butterfield, also making his first start after recovering from a knee injury following a summer move from Oakwell, tried his luck from the edge of the box, but Woods got down to make a smart save.
Doncaster, though, were unfortunate not to level following a goalmouth scramble 10 minutes before half-time, Shelton Martis seeing his goalbound effort blocked in the melee.
At the other end, Rovers captain Jamie McCombe somehow blocked Elliott Ward’s bullet header from point-blank range in first-half stoppage time.
Norwich were forced into a change on 50 minutes after Kane fell awkwardly on his ankle, and had to be taken from the pitch on a stretcher.
Woods made a smart double save to parry Holt’s low effort from 20 yards and then beat away the follow-up effort from substitute Simeon Jackson.
Holt went close again with a diving header at the far post on 72 minutes, before then combining with Jackson as the Canada frontman skipped around the goalkeeper but could only slide the ball back into the side-netting.
Doncaster almost snatched an equaliser with two minutes left after full-back James Husband darted down the left and his low centre was stabbed goalward by substitute David Syers, when better contact would have seen the ball in the net rather than the grateful hands of Bunn.
Norwich, though, did enough to secure safe passage into round four for the first time since the 1995/96 season, as they paid a fitting tribute to Bond, who guided the club to Wembley in this competition almost 40 years ago.
Norwich: Bunn, Elliott Bennett, Ward, Ryan Bennett, Tierney, Butterfield, Fox, Chris Martin (Hoolahan 71), Tettey, Holt, Kane (Jackson 50). Unused substitutes: Rudd, Turner, Morison, Barnett, Murphy.
Doncaster: Gary Woods, Quinn, Martis, McCombe, Spurr (Bennett 61), Cotterill, Martin Woods (Syers 50), Harper, Husband, Blake (Brown 57), Hume. Unused substitutes: Maxted, Rob Jones, Middleton, Keegan.
Referee: Darren Deadman (Cambridgeshire).
Nuri Sahin’s first goals for the club kept Liverpool’s grip on the Capital One Cup as Brendan Rodgers’ young lions came from behind to win 2-1 at West Brom.