Johnson beat off competition from fellow nominees Simon Grayson of Blackpool, Martin Foyle of Port Vale and Kevin Bond of Bournemouth to land the award.
City won five of their seven games in March to consolidate the team's charge for promotion to the Championship.
The Robins are four points clear of third-placed Nottingham Forest with just six games remaining.
Johnson told Sky Sports News: "It's nice when you get the manager of the month this late in the season, because it means your team have accumulated quite a few points."
Johnson revealed his team are still hopeful of catching leaders Scunthorpe.
"We've still got Scunthorpe in our sights but I suppose we're looking over our shoulder more if the truth be known," he added.
"We're trying to keep positive and look ahead and make sure we keep it in our hands.
"It was a great accumulation of points [in March] and it really did put us in a strong position."
Sky Sports