Rodgers' side opened last month with a 5-0 win at Hearts that clinched the title - the club's sixth successive top-flight success.
And they finished April with their biggest Ibrox victory over Rangers, winning 5-1.
Celtic remain unbeaten domestically this season.
And they can complete a treble if they win the Scottish Cup final against Aberdeen on 27 May, having beaten the Dons in November's League Cup final.
Rodgers also won the August, October and December manager of the month awards.
"We obviously had some great results and even more important was how we played," said Rodgers after receiving his award.
"It is always good to win, but it is even better to win well.
"We had some outstanding performances and the team and the staff, we accept the award on behalf of everyone at the club."
Brendan Rodgers is the Scottish Premiership manager of the month for April after guiding Celtic to three wins and two draws.