Nathan Broadhead put Town in front but Jonathan Rowe equalised to leave the scoreline 1-1 between the Blues and Norwich at half-time.
Town boss Kieran McKenna made four changes from the team which won 2-1 at Watford on Tuesday.
Harry Clarke returned at right-back with fellow local boy Luke Woolfenden at right centre-half, while Wes Burns and Nathan Broadhead are back in the three behind striker George Hirst.
Former Canaries loanee Brandon Williams, Axel Tuanzebe, Omari Hutchinson and Marcus Harness dropped to the bench having started at Vicarage Road.
Norwich made two changes from the team which beat Sheffield Wednesday 3-1 on Wednesday with Jack Stacey and Sam McCallum taking over from Kellen Fisher, who is among the subs, and Dimitrios Giannoulis, who missed out on a place in the 20. Former Blues academy midfielder Liam Gibbs was on the bench.
Town, with their fans in fine voice both before and after kick-off, dominated the early stages with the Canaries making their intentions to sit back and frustrate clear from the off.
The Blues threatened for the first time in the seventh minute when Cameron Burgess nodded a Sam Morsy free-kick from the right over the bar.
On nine, Norwich striker Ashley Barnes was very fortunate to avoid a yellow card for going through the back of Town keeper Vaclav Hladky with the ball already out of play. Referee Josh Smith felt a talking-to was sufficient.
A minute later, Borja Sainz jumped into Clarke as they went for an aerial ball and rolled around on the ground holding his face. However, referee Smith had already awarded a free-kick to Town and clearly wasn’t fooled in the slightest.
Town should have gone ahead in the 20th minute. As Norwich started to show some attacking intentions for the first time, the Blues picked their way through them, Burns initially hitting a low cross-shot which was blocked.
The ball came back to Burns and was played inside, before Broadhead skipped his way past defenders into the area and had only Angus Gunn in the Norwich goal to beat but clipped the ball wide of the far post with most of Portman Road already starting their celebrations.
Town continued to look a threat and a minute later, Burns crossed deep from the right to Leif Davis breaking on the left of the box from where the full-back struck a powerful shot which was straight at Gunn.
On 23, Broadhead had another chance to put the Blues in front. Burns again supplied the cross from the right but slightly behind the Welshman, who dug it out and struck an effort which flew just past the same post.
Five minutes later, Town had another great chance to go in front. Norwich were caught in possession trying to play it out from the back and Hirst fed Burns in space to his right on the edge of the box but the Wales international smashed his shot well over.
Norwich hit their first effort of the game in the 29th minute, Marcelino Nunez’s effort deflecting wide off a Town player after a corner had been cleared.
From the subsequent flag-kick, the ball reached Sam McCallum in space on the right of the box but his low shot - which looked on target - was blocked. A minute later, Sainz hit over from 20 yards.
The Canaries had started to threaten for the first time, but in the 34th minute the Blues finally scored the goal their overall superiority had deserved.
Town broke following a Norwich corner through Chaplin, who did brilliantly to beat his man and feed Hirst bursting into the right of the box. Shane Duffy stabbed it out for the Blues’ first corner of the game, from which they went in front.
The ball came in from the right, it was headed out, Luongo nodded back in from the edge of the box, Hirst knocked down to Broadhead, who this time slammed his opportunity into the net to send Portman Road wild.
But Town’s lead would last only five minutes. The ball was sent back in from the right after a cross from the left had flashed across the box and Jonathan Rowe struck an overhead kick against Woolfenden. The ball fell back to the Norwich forward, who shot past Hladky, to send the away fans into raptures. Video replays suggested Rowe might have been offside as the cross came back in.
The Blues set about restoring their lead in what remained of the half and in the first of three added on minutes, Hirst chipped a ball forward into the area for Broadhead, but the Welshman failed to make a decent contact under pressure and scuffed wide.
That was the last chance of a first half in which the Blues had done enough to have been well ahead prior to Norwich’s leveller.
After the game got off to a slowish start, Town took control against a Norwich side happy to sit back.
The Blues created a number of chances from which they should have taken the lead before Broadhead took the third of three which had fallen to him.
At the other end, Norwich had created little but some hesitant defending in the Blues backline allowed the Canaries their equaliser.
However, if Town continue to play as they have in the first half, more opportunities are likely to come before the end.
Town: Hladky, Clarke, Woolfenden, Burgess, Davis, Morsy (c), Luongo, Burns, Chaplin, Broadhead, Hirst. Subs: Walton, Tuanzebe, Williams, Taylor, Harness, Hutchinson, Jackson, Scarlett, Ladapo.
Norwich: Gunn, Stacey, Duffy, McLean (c), McCallum, Sara, Nuñez, Rowe, Sainz, Barnes, Idah. Subs: Long, Gibbs, Fassnacht, Placheta, Batth, Hernandez, Forshaw, Fisher, Springett. Referee: Josh Smith (Bedfordshire).