What was the first aircraft that could fly inverted?
Deliberate momentarily inverted flight
The first reported deliberately inverted flight was probably in a Blériot model XI.


Inverted flight was also being considered. Several pilots had inadvertently found themselves upside down as a result of wind gusts, but no one had yet attempted it intentionally. Adolphe Pegoud decided he would be the one to try.
Pegoud was one of the first true aerobatic pilots. An accomplished test pilot with the Bleriot team, he was also the first flyer to jump from a plane with a parachute. Both Pegoud and the aircraft landed, unharmed. Having decided to try upside-down flying, Pegoud first practiced in a hanger [sic] in a plane hung upside down from the ceiling. He correctly assumed the controls would have to be operated in a reverse manner and he wanted to get the feel of it before actually flying.
It must have worked. Shortly thereafter, Pegoud made the first public demonstration of inverted flight in September of 1913. During this flight, he was the first to perform a half roll to an inverted position--another aerobatic maneuver was born. The flight did point out one potential problem: While Pegoud was prepared to fly inverted, the aircraft was not. It proceeded to drench the flyer with fuel!

From The history of aerobatics
Deliberate sustained inverted flight
This would have to have been an aerobatic aircraft specially modified for inverted flight

flop tubes in the fuel tank?
fuel-injection not carburettor?
strengthened and/or braced to allow wings to support fuselage in inverted position?

1925
June 1925, No 32 Squadron did an air display demonstrating Flight-converging bombing at the R.A.F Display, Hendon. Scott was selected to do individual aerobatics in a brand-new Snipe which he was allowed to paint red, this pleased Scott greatly as it meant that he was also allowed to practice his aerobatics at a low altitude, rather than above 2000 feet which was R.A.F regulations at that time. In Scott's book he tells of how for his solo display he was allotted exactly seven minutes during the luncheon break, to complete his show but after just two minutes, a flying wire broke in the near edge of the port side, anxious not to cut his allotted seven minutes down too much he began to fly the aircraft upside down in an effort to reduce the strain on the flying wires, he continued flying in an inverted position for some time until he noticed a worrying quiver in the top plane and promptly landed slightly short of his seven minutes.
From Wikipedia
1933 record
In 1933, NC872H, a Boeing 100, was flown inverted for 4 hours 5 minutes and 22 seconds.


After its career with Pratt &amp; Whitney, the Museum of Flight's aircraft was sold to Milo Burcham on September 27, 1933. He modified it for skywriting and air-show work, painted it blood red, added fuel and oil systems for inverted flight, and flew it upside down from Los Angeles to San Diego.

From Seattle Museum of Flight