Rovers had the game's best early chances, Michael O'Halloran heading straight at Crawley goalkeeper Paul Jones before loan debutant Amoo pulled his low shot wide on 15 minutes.
But the former Liverpool youngster - who agreed a deal with Tranmere until the end of the season after being released by Preston this week - marked his debut with the opening goal, heading in from close range after Jones parried his own defender Mark Connolly's header off the line.
The visitors were unfortunate not to be level seven minutes later after Billy Clarke's follow up to Mike Jones' saved shot hit the post and rolled out.
In bizarre circumstances, Bakayogo doubled Rovers' lead six minutes before half-time as his looping left-wing cross bounced off the far post and went in off keeper Jones.
Defender Ash Taylor almost put the hosts three up immediately after the break but his right-footed strike from the edge of the area rebounded back off the crossbar.
Crawley - who were largely restricted to long-range efforts - saw Nicky Adams' 25-yard free-kick flash wide after 50 minutes.
The game started to fade away although Amoo and substitute Jean-Louis Akpa Akpro continued to cause the Crawley defence problems.
Rovers keeper Owain Fon Williams parried a late long-ranger from Clarke to safety as Crawley's challenge petered out.