The 38-year-old BJP leader defeated Congress nominee Aleixo Reginaldo Lourenco.
Sawant secured 20 votes while Lourenco got 15.
When pro-tem Speaker Siddharth Kuncolienkar called for the election, BJP, Maharashtrawadi Gomantak Party (MGP), Goa Forward Party and Independent MLA Govind Gawade moved separate motions supporting Sawant.
On the other hand, Congress legislators Luizinho Faleiro, Digambar Kamat, Isidore Fernandes moved the motions supporting candidature of Lourenco.
The pro-tem Speaker then called for the head count of those supporting the motion in favour of Sawant following which 20 MLAs from the ruling benches stood up.
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After Sawant won, Kuncolienkar invited him to occupy the Speaker's chair.
With the resignation of Congress MLA Vishwajit Rane last week, the effective strength of the 40-member House is now reduced to 39.
Out of the 39 members, 20 from BJP and its alliance partners voted in favour of Sawant. The pro-tem Speaker did not vote. The Congress nominee received 15 votes.
NCP legislator Churchill Alemao, who had supported the BJP-led alliance during the trust vote in the Assembly last week, remained seated when the pro-tem Speaker called for the head count.
Chief Minister Manohar Parrikar later said in the House that the two legislators were stuck in traffic.
Parrikar congratulated Sawant on being elected as the Speaker and said he is an experienced legislator who has actively participated in various debates in the House.
The Chief Minister said the House is expected to have healthy debates in future which will contribute towards bringing better legislations.
Leader of Opposition Chandrakant Kavlekar said he expects that the Speaker would be impartial in doing his job.
The three-day session of the Assembly began today.
State Governor Mridula Sinha will address the House tomorrow and Parrikar will present the state Budget on March 24.