Sjöstedt, a former metal worker and EU parliamentarian, has always been the strong favourite in the four-way leadership contest with Ulla Andersson, Hans Linde and Rossana Dinamarca.
"I am delighted and proud. It is the finest political job I can think of," said Sjöstedt to news agency TT.
He pointed out that he had not got the job yet and had to be confirmed in a vote at the party congress in January.
He also said that he was open for a dual leadership of the party.
Leadership candidate Hans Linde said to TT that he respects the work of the nominations committee and believes that Jonas Sjöstedt will be a good party leader for the Left Party.
"I have decided to withdraw my candidacy for the leadership post."
Fellow candidate Rossana Dinamarca said she also respected the nomination choice but would be not withdrawing her name from the leadership race.
"I was a candidate yesterday and I will be a candidate today also. This has been a good process which has been open and I think that it would be a shame to end it here and now. Jonas is a candidate that I respect a lot but I also think that I would be a very good party leader" Dinamarca said to TT.
Since 2010, Sjöstedt has served as an MP in the Swedish Riksdag, representing the northern Swedish county of Västerbotten. Before that, he followed his wife, who was a Swedish representative to the United Nations, to New York. There, he lived in Brooklyn for a time, and was active in the Socialist Party there. He has also worked for Volvo Trucks.