The first ever goal in international football was scored by William Stanley Kenyon-Slaney of the Wanderers for England against Scotland on 8.3.1873. It came in the first minute of the match, which was the second ever international.
Scotland's first scorer was Henry Waugh Renny-Tailyour (Royal Engineers) in the 25th minute of the same match, which England won 4-2.
Wales had to wait until their fourth international before registering a goal. Their first score was made by William Henry Davies (Oswestry) in the 2-1 defeat to England on 18.1.1879.
Samuel Johnston (Distillery) got the first international goal for Ireland as they went down 7-1 to Wales on 25.2.1882.
Carlos Dickinson (Belgrano) registered Argentina's first international goal in a 6-0 defeat of Uruguay, 20.7.1902.
Josef Taurer (Wiener AC) scored Austria's first goal in a 5-0 win over Hungary on 12.10.1902.
József Pokorny (Hungary)
In their second international on 11.6.1903 Hungary avenged a 5-0 defeat at the hands of neighbours Austria with a 3-2 victory. József Pokorny (Ferencváros) got the first.
Carlos Céspedes (Uruguay)
Uruguay waited 14 months for their revenge over Argentina- Carlos Céspedes (Nacional) scored the opener in their 3-2 victory on 13.09.1903. Germán Arímalo and Carve Urioste (Nacional) had both scored in Uruguay's first match, but these were own goals!
Belgium v France 1.5.1904. in the 7th minute Georges Quéritet (Racing Brussells) opened the scoring for Belgium in their 3-3 draw with France. 5 minutes later Louis Mesnier (Cercle Athlétique de Paris) became France's first international goalscorer.
Eddy de Neve (Netherlands)
Netherlands first international outing was a a triumph for Eddy de Neve (Velocitas Breda). The Coupe van den Abeele was at stake so after 1-1 at 90 minutes extra time was played. De Neve got all four goals as Netherlands beat Belgium 4-1.
Bohemia (Czechoslovakia) played Hungary on 1.4.1906 and Jindřich Valášek (Meteor VIII) scored the opening goal in a 1-1 draw.
In their second international on 8.3.1908 Adolf Frenken (Winterthur) gave Switzerland the lead against France, but they lost 2-1.
Fritz Becker (Germany)
Germany- Fritz Becker (Frankfurter Kickers) got Germany's first international goal as they were beaten 5-3 by Switzerland on 5.4.1908.
Karl Gustafsson ( Sweden)
Denmark- Nils Middelboe (Kjøbenhavns Boldklub) got the first of Denmark's goals in a 9-0 win over France on 19.10.1908.
Pietro Lana (Italy)
In their first international, versus Argentina on 27.5.1910, Chile took an unexpected 11th minute lead through Frank Simmons (Badminton). Argentina came back to win 3-1.
22.10.1911 saw Uno Lindbäck (HIFK) gave Finland the lead in the 3rd minute of the match against Sweden. Sweden eventually won 5-2.
On 29.10.1911 Luxembourg's Albert Elter (Racing Club Luxembourg) gave his country a 1-0 lead over France France came back to win 4-1.
Vasily Butusov (Russia )
Rubens Salles (Brazil)
In their second international on 27.9.1914 Brazil secured a great 1-0 victory away to Argentina, with Rubens Salles (Paulistano) getting the winner in the 13th minute.
Tommy Swords (USA)
USA captain Tommy Swords (Fall River Rovers) got the opening goal in the 3-2 win over Sweden on 17.8.1916.
Haruyoshi Kitamura (Tokyo Normal Higher School ) scored both Japan's goals in their 15-2 loss to the Philippines.*
On 11.5.1919 Paraguay lost 5-1 to Argentina in their first ever international. Faustino Casado- 'Casadito' (Nacional FBC Asunción) opened their international account with a 30th minute equalizer.
Hassan Allouba (Egypt)
Egypt's Hassan Allouba (Tersana/ Arsenal Sporting Club Cairo) got the first African international goal as his side went down 2-1 to Italy on 28.8.1920.
Patricio Arabolaza Aramburu (Spain)
On the same day (28.8.1920) Patricio (Real Unión) got the goal as Spain began international football with a 1-0 win over Denmark.
Yugoslavia- in their second international on 3.9.1920 Artur Dubravčić (Concordia Zagreb) got Yugoslavia's first international goal as they were beaten 4-2 by Egypt.
* Note: I have been unable to identify Philippines (1913) first goalscorer as yet, and China do not recognize their participations in the Far Eastern Games as being full internationals.