The Brazilian bent in a fine effort following a corner in a first half which could have seen the visitors home and dry, but the Clarets were twice denied by fine saves from goalkeeper Ederson.
But, with the game inside the final eight minutes, Gudmundsson broke behind the City back line to volley in a crashing equaliser which the keeper couldn't keep out.
Burnley will be buoyed by the result despite seeing their winless run stretch to ten, while City were denied a fifth successive win in all competitions.
After an initially nervy start in what was a hostile home atmosphere, the visitors soon settled into their rhythm and carved out a couple of early chances.
Vincent Kompany was inches away from meeting a Bernardo Silva cross at the back post following a corner, and it was from such a set piece that the opening goal came.
Danilo bends in classy opener
After a clever short routine was cleared behind, City took a second corner short to Silva on the left. The Portuguese held onto the ball unchallenged and found Danilo in space, with the full-back having time to check onto his right foot and curl the ball into the far top corner.
Burnley were not without chances of their own. Their forwards picked their moments to press the City back line well, but it was from a hopeful dink into the area that their first opportunity came. Ben Mee was left alone in the box as the ball dropped and he volleyed towards the bottom right, but Ederson got down quickly to make a strong save. Soon after, Mee headed wide from a corner.
City had enough chances in the first half to make themselves comfortable, but Nick Pope denied Kevin de Bruyne and Raheem Sterling while Sergio Aguero was twice thwarted by last-ditch Burnley defending.
Burnley go close in second half
The second half brought more City pressure but Burnley responded well, carving out opportunities of their own.
Sterling was the most guilty of missing chances, but he may suggest Aguero's pass was a little behind him when he hooked a shot over what was essentially an open goal early in the half.
On another day, Mee might have been celebrating a hat-trick soon after the hour mark. Gudmundsson fired in an inviting free kick following a foul by Nicolas Otamendi, but the cross was just out of reach for the straining Burnley captain.
Soon after, Lennon must have thought he had scored on his home debut only to be denied by a stunning save from Ederson, who tipped his crashing shot against the woodwork.
At the other end, though, it was proving to be an off day for Sterling, who submitted a strong contender for the miss of the season.
A yard out, unmarked and with half the goal unguarded, he scuffed his finish and sank to the ground as he watched it bobble wide. He was substituted soon after.
He would regret the miss as Burnley pulled themselves level with time looking to be against them. Substitute Matthew Lowton floated the ball temptingly for Gudmundsson, who broke beyond the City defence and fired a powerful body across the body of the goalkeeper and inside the far post.
Burnley baffled by City set pieces
Given the contrasting styles of the two sides it was not obvious that City would threaten so regularly from corners, but Burnley seemed unable to cope with their varied routines throughout.
Moments before Danilo's opener, a corner had been cleverly knocked straight towards goal for Gundogan to chase, and his pass brought about the corner from which the first goal was scored.
The Germany international was a constant problem for the home defence from such situations, picking up pockets of space seemingly at will as Guardiola's side offered the taker multiple short options. It was clever use of the set pieces from a side at a clear height disadvantage.
Hosts rue wrong man in right places
Ben Mee is in his seventh season at Burnley and the defender has six league goals to his name, so it is perhaps unsurprising that he didn't take any of the chances that came his way.
Two chances from set pieces went begging, but it was his volley in the first half that Sean Dyche would have rather seen fall to Sam Vokes or Ashley Barnes.
It was a decent enough effort, but a little more power or more incisive placement would have left Ederson with no chance. Mee gave the keeper a sliver of hope, and the Brazilian responded with a fine save.