Everton striker Lukaku scored four goals in three Premier League matches in March, becoming the club's first player to hit the 20-goal mark in the Premier League era.
"I'm really happy," Lukaku said. "It gives me another boost. I'm playing well at the minute and the team is doing really well."
Lukaku was chosen as the month's best player by a panel of Premier League captains and supporters. The other five contenders were Liverpool's Gini Wijnaldum, Crystal Palace's Mamadou Sakho and Nathan Redmond of Southampton, as well as Bournemouth duo Josh King and Artur Boruc.
Bournemouth boss Howe, meanwhile, was named the month's best manager after the Cherries were unbeaten in March.
With the club teetering just above the relegation zone at the end of February, they drew 1-1 at Manchester United while winning against West Ham and Swansea.
Crystal Palace winger Andros Townsend's virtuoso goal against West Brom was voted the goal of the month.