The 49-year-old led the Toffees to three successive wins during the month to maintain their challenge for a top-four finish.
Moyes beat Manchester United boss Sir Alex Ferguson, Southampton manager Mauricio Pochettino and Fulham's Martin Jol to the award.
It is the 10th time Moyes has won the Manager of the Month award.
His Everton side are currently sixth in the Premier League table, four points behind fourth-placed Chelsea.
Tottenham defender Vertonghen was named Barclays Player of the Month ahead of Liverpool striker Luis Suarez, Fulham forward Dimitar Berbatov and Manchester United goalkeeper David De Gea.
Vertonghen, 25, was an ever-present during Spurs' four Premier League games in March, scoring three goals.
The Belgian international is enjoying an impressive debut season in England, having moved to White Lane from Dutch club Ajax last summer.