In just his second appearance since linking up with the Edinburgh club, Socceroos midfielder Irvine spurned a great chance to give his side the lead in the first half, sending his shot from a metre out flying over the cross bar.
Alfredo Morelos made the home side pay, firing Rangers into a 1-0 lead early in the second period.
Irvine had another, albeit trickier, opportunity to level the scores, but he sent his thundering header inches wide of the Rangers goal.
Morelos was fortunate to avoid being sent off in the first half after appearing to stamp on Ryan Porteous.
But referee Kevin Clancy missed the extent of the incident, leaving Morelos to add insult to injury for Hibs with a clinical winner.
Steven Gerrard's side needs just eight more wins to clinch its first Premiership crown in a decade.
Second placed Celtic are 23 points adrift of its Glasgow rival after a 2-0 victory against Hamilton at Parkhead.
After four Scottish Premiership games without a win for the first time since 2000, champion Celtic badly needed three points.
Leigh Griffiths struck from long range on 11 minutes and Odsonne Edouard added a neat second early in the second half.
Livingston secured a 2-0 triumph over Kilmarnock to extend its unbeaten run in all competitions to 12 matches.
Jaze Kabia notched his first goal for the Lions in the 89th minute and Scott Pittman, celebrating his 250th appearance for the club, bagged the second in stoppage time.
St Mirren crushed Dundee United 5-1 at Tannadice with goals from Joe Shaughnessy, Jamie McGrath, who scored two penalties, Dylan Connolly and Kristian Dennis.
Motherwell came from behind for a 2-1 win at fellow strugglers Ross County, while Aberdeen drew 0-0 with St Johnstone.