Goals from Gabriel Sara and Josh Sargent were not enough to secure three points against Swansea City in the Canaries' final home game of the 2023/24 season.
David Wagner made one change to his starting 11 that drew against Bristol City last weekend, bringing in Christian Fassnacht for Ashley Barnes.
Club captain Grant Hanley also returned to the bench for the first time since picking up an injury last month.
The opening stages of the game saw a scramble in the Swansea box with chances falling to Sargent, Sara and Borja Sainz in the space of a few seconds, but none could find the back of the net.
An early opportunity came for the Swans through Liam Cullen, who had a shot smothered by Angus Gunn and cleared by Jack Stacey on the rebound.
A lovely Sara turn and and manoeuvre got a cross to Fassnacht in the box, but his effort was defected away for a corner after 15 minutes.
It was the visitors who would find the breakthrough first, however, as Jamie Paterson found himself space in the box to shoot on the left and fired into the top left corner to make it 1-0.
Quick reactions from Carl Rushworth in goal for Swansea denied Sargent again at close-range just before the half-hour mark, after a brilliant run from Marcelino Núñez on the edge of the area.
It didn't take long for the Canaries to find an equaliser, when Fassnacht perfectly placed a header to Sara going forward, and the Brazilian powered a shot onto the cross bar and into the back of the net.
Sara then turned from scorer to provider, as seconds before the half-time whistle he picked out Sargent in the box, who headed the ball into the back of the net to put Norwich in front.
The second half opened with Norwich on the attack, as Fassnacht was involved again to set-up Sargent in the box, but his effort on target was blocked by a pink shirt.
Fassnacht had his own chance shortly after to head home but couldn't nod his effort the right side of the post.
In the 51st minute, the visitors were awarded a penalty after Gunn looked to bring down a Swansea player in the box. Captain Matt Grimes stepped up and converted from the spot.
A header from Shane Duffy was cleared off the line by Jamal Lowe in the Canaries' best chance with 10 minutes left on the clock.
Swansea had the chance to put themselves ahead again with just three minutes of normal time remaining, but Gunn made a great save to deny Lowe.
The final effort from the home side to take all three points came from Núñez who could only poke a shot wide of the target.
Norwich travel to Birmingham City next on Saturday, May 4 with a place in the play-offs still in their own hands.
Norwich City: Gunn; Stacey, Duffy, Gibson, McCallum (van Hooijdonk 90+5); McLean (C), Núñez, Fassnacht; (Rowe 67) Sara, Sainz (Barnes 67, (Sørensen 82)), Sargent.
Unused subs: Long, Hanley, Batth, Fisher, Montóia.
Goals: Sara (34), Sargent (45+3)
Bookings: McCallum (77)
Swansea City: Rushworth; Key, Cabango, Darling (Wood 76), Tymon; Fulton, Grimes (C), Paterson (Lowe 67); Ronald, Cullen (Yates 67), Cooper (Walsh, 82).
Unused subs: Fisher, Patino, Kuharevich, Humphreys, Aboulai.
Goals: Paterson (24), Grimes (53 pen)
Bookings: Paterson (45), Cullen (57)
HT: 2-1
FT: 2-2
Referee: James Linington
Venue: Carrow Road
Attendance: 26,429