Swansea gave an improved display in the 3-1 defeat by Manchester City, but are one point above the relegation zone.
Ryan Giggs and Gianfranco Zola have been linked with the Swansea job, but Amat believes things are improving.
"We believe in him and our ideas so we have to keep playing like this and the results will come," he said.
"All the team believe in Francesco, we agree with him all the time, we are working so hard and we showed the crowd we can play very good football against one of the best teams in the Premier League."
Swansea's American owners, Jason Levien and Steve Kaplan, have been alarmed by the Swans' recent poor form.
With difficult fixtures against Liverpool on Saturday and Arsenal to come, however, it is understood the quality of Swansea's next few performances will have as much bearing on Guidolin's future as the results.
Having been unimpressive in defeat at Southampton, there was a marked improvement in Saturday's encounter with Premier League leaders City.
Former Udinese boss Guidolin joined Swansea in January as head coach and was given a two-year contract in May.
Amat says the Swans now need luck to go with their improved form.
"Against Manchester City the performance was good, but we need some luck as well," he said.
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Swansea City defender Jordi Amat says the players "believe" in manager Francesco Guidolin despite winning only one Premier League game this season.