"Special night, special game, special shirt and a goal," Suarez wrote on social media alongside a picture of him holding his shirt from the match.
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"What more could I ask for in unique and unforgettable moments for my country?"
The South American Football Confederation said Suarez's 29 goals came in 62 games, two games more than Messi, and his latest was typically spectacular, a bicycle kick from four meters out with 11 minutes remaining.
Federico Valverde doubled Uruguay's lead in the final moments of the match to cement a remarkable turnaround in fortunes for the side under Diego Alonso.
Alonso took over from Oscar Tabarez in December last year after four successive defeats put their qualification hopes in jeopardy and he has since guided his new charges to four straight wins.
Uruguay finished third in the qualifying group behind Brazil and Argentina and ahead of Ecuador, who took the fourth automatic qualifying place.