The 24-year old became a father for the second time on Monday - his wife gave birth to a baby girl - and he scored the only goal of the game to secure back-to-back wins for his team.
The Robins, who had won only six league games all season going into the fixture, secured the win with a goal after only three minutes.
Tranmere were caught cold when Ian Westlake's corner was cleared to Bird on the edge of the penalty area. Bird controlled the ball with his right foot and, as it bounced up, he thumped a shot with his left into the top right-hand corner of goalkeeper Danny Coyne's net.
For much of the first half Tranmere looked a shadow of the side that beat league leaders Leicester City last week and it wasn't until just before half-time that they created a chance of any note.
A free-kick taken by Charlie Barnett spun into the air off a defender and, as the ball dropped, right-back Ryan Shotton attempted a spectacular overhead kick that hit the foot of a post.
The second half followed much the same pattern with hard-working Cheltenham getting bodies behind the ball and using the pace of their quick front players on the break.
On the rare occasions Tranmere did break through they were unable to convert their chances.
The best ones fell to left-back Andy Taylor, whose shot from a tight angle was touched over the bar by goalkeeper Scott Brown, and midfielder Antony Kay, whose header from a Ryan Shotton cross brushed the crossbar.
Cheltenham's effervescent manager Martin Allen will still believe his team have a slight hope of avoiding relegation despite the ten-point gap between them and safety.
For Tranmere this was an opportunity missed to cement a place within the play-off zone and boss Ronnie Moore will remain vexed by his team's poor away form given that they have the best home record in the division.