Rangers took time to get going at the Victoria Stadium, with all four of its goals against the Gibraltarian minnow coming in the second half.
Liverpool loanee Ojo curled the ball into the far corner six minutes after Ryan Jack opened the scoring, before Connor Goldson added a third from close range in the 68th minute.
A late fourth goal was provided by substitute Alfredo Morelos as Steven Gerrard's side took complete control of the tie ahead of the return match at Ibrox on 18 Thursday.
Gerrard's men eased to a 5-0 win in their friendly with Oxford United but they struggled to get going in a first half devoid of any real action against St Joseph's.
But it took just five minutes of the second half for Rangers to create its first chance of the match and Jack made the most of it by sending his side-footed volley past Francisco Mateo Vera.
Ojo's debut strike followed soon after as he picked his spot from 15 yards to cap an impressive first competitive outing for his new club.
St Joseph's struggled to maintain its tempo and Goldson killed off the game when tapping in from close range after Borna Barisic's free-kick was scooped off the line by Mateo Vera.
Morelos rounded off the scoring with 13 minutes remaining, bundling over after seeing his initial effort – which may have crossed the line – pushed out by Mateo Vera and Goldson's follow-up from inside the six-yard box rattle the crossbar.