It was Bridges who put the Tigers on the way to victory after debutant Tom Pope was penalised for handling Michael Turner's cross, although the former Sunderland and Leeds striker had his spot-kick saved by Ben Williams.
He followed up to finish the rebound two minutes before the interval.
It made up for an earlier miss by Bridges when he half-volleyed tamely at Williams after a searching cross from Stuart Elliott had found him unmarked at the back post.
But Crewe had their chances as well. Shaun Miller should have done better when he was released by Pope, but Hull reserve keeper Matt Duke got his hand to the youngster's rising drive.
Young Crewe winger Byron Moore was the star for the home side as he turned neatly inside the area straight after Bridges' opener but was denied by a reaction save from Duke.
Miller also shot at the keeper from a tighter angle as Crewe threatened an equaliser before and after the break.
Gary Roberts was denied by Wayne Brown's sliding intervention and the visitors made the Railwaymen pay shortly after that.
A neat interchange between Bridges and Nicky Featherstone released Garcia, who strode into acres of space before unleashing a 25-yard drive high into the top corner in the 56th minute.
Hull rubbed in their superiority in terms of their clinical finishing at least with a third goal with 20 minutes left.
A misunderstanding between Chris McCready and keeper Williams allowed McPhee to wriggle in and flick the ball into an unguarded net.
The former Port Vale striker came close to netting his side's fourth but messed up his header from six yards out.