There was one change in the Swansea side with Jason Scotland replacing Gorka Pintado who was on the bench.
A foul by Rangers' skipper Martin Rowlands on Garcia Jordi Gomez in the seventh minute gave Swansea their first sight of goal, but Gomez sent his free-kick whistling over Radek Cerny's crossbar.
Five minutes later, Gary Monk sent Scotland racing clear but the Trinidad & Tobago striker slipped as he lined up his shot.
The Swans suffered a blow in the 24th minute when Dorus de Vries was injured in a collision with Rowlands.
The Dutchman was stretchered off and, with no goalkeeper on the bench, Tate took over between the sticks.
Swansea took the game to Rangers and forced a number of corners before a Ferrie Bodde flick just flew wide.
The half drew to a close with the Swans well on top while stand-in keeper Tate had not even been troubled in the home goals.
In the opening three minutes of the second half Swansea peppered the visitors' goal with shots from Bodde and Darren Pratley.
Swansea's play through the middle was superb with Leon Britton, Darren Pratley and Bodde ruling the centre of midfield, but they just could not carve out any clear openings.
Rangers skipper Rowlands was replaced on the hour by Daniel Parejo as Iain Dowie tried to spark some life into his side.
It was not until the 70th minute that Tate was forced into his first save after punching an Akos Buzsaky free-kick clear.
Parejo hacked a Pratley shot off the line as the Swans camped in the Rangers half and then Britton was stopped in full flight by Damion Stewart, who earned himself the R's fifth yellow card of the evening.
Rangers seemed content to hold out for a draw which was surprising as the Swans were playing with a substitute keeper, but there was no urgency in their play at all and Swansea were the only side looking for victory.