The Gers were looking to recover from last week's Scottish League Cup final loss to arch Glasgow rivals Celtic in manager Michael Beale's first defeat since taking charge at Ibrox in December.
They took an early lead through defender Connor Goldson's sixth-minute goal and were 2-0 up before the half-hour mark thanks to a powerful shot from Fashion Sakala.
Rangers captain James Tavernier appeared to have made the game safe before half-time when he scored from the penalty spot after VAR detected a handball inside the box.
But the second half was a more nervous affair with Kilmarnock defender Jeriel Dorsett pulling a goal back on the hour mark as the hosts struggled to replicate their first-half dominance.
Even so the win saw Rangers narrow the gap ahead of reigning champions' Celtic's match away to St Mirren on Sunday.
"You can only beat the team in front of you," said Beale. "We will roll on again to Hibs on Wednesday.
"Hibs will come for us and we'll look forward to that as we'll have more spaces to play."
Hearts push on
Josh Ginnelly scored twice as Hearts strengthened their grip on third place with a 3-0 win at home St Johnstone.
The Englishman struck in each half before Jorge Grant put the result beyond doubt for Hearts, who remain five points clear of Edinburgh rivals Hibernian in the race for a European place.
Motherwell moved further away from the relegation zone with a 2-0 win away to Ross County, who finished the match with 10 men following Jack Baldwin's 70th-minute red card.
Baldwin was sent off for bringing down Kevin van Veen, who then went on to score twice for the Steelmen.
Hibernian recovered from falling a goal behind to win 4-1 at Livingston after Joel Nouble opened the scoring in the ninth minute.
Elie Youan scored twice before half-time for Hibs, with Jack Fitzwater then sent off for Livingston.
Stephane Omeonga's own goal made it 3-1 to Hibernian before Mykola Kukharevych added a fourth deep into stoppage time.
Aberdeen ensured former manager Jim Goodwin's first match in charge of bottom club Dundee United ended in a 3-1 home defeat.
Goodwin, just over a month since being sacked by Aberdeen following a 6-0 thrashing by Hibs that capped a woeful run of results, saw Luis Lopes backheel his former club in front shortly after half-time.
Jamie McGrath equalised from the penalty spot to give the home side hope, but two goals inside the final 10 minutes from Ross McCrorie and Marley Watkins left United four points adrift of Kilmarnock and Ross County.
Aberdeen's first win at Tannadice in seven years kept them within two points of fifth-placed Hibernian.
