Steve Cooper's side kept pace with the teams at the top of a table thanks to an added-time penalty against Stoke City on Wednesday night, while Middlesbrough got back to winning ways by beating Coventry City 2-1 on Tuesday.
The side from Wales travelled to Stoke looking to keep up with the top three last time out, but they were dealt a bad start when Nick Powell put the hosts ahead after just six minutes, before right-back Connor Roberts equalised for the Swans soon after.
The game looked set to finish 1-1 until Jack Clarke tripped Kyle Naughton in the box in the sixth minute of added time, resulting in a penalty which Andre Ayew coolly dispatched to seal all three points for Cooper's men with the final kick of the match.
That result followed a disappointing 3-1 loss to Bristol City, when Ayew gave Swansea the lead with another penalty before Nahki Wells, Kasey Palmer and Antoine Semenyo turned the game around and condemned the side from South Wales to their sixth league defeat of the campaign.
However, because of that last victory, the Swans now sit just one point behind second-placed Brentford with two games in hand, one of which takes place on Tuesday night against Blackburn Rovers.
Their strong league position is largely down to the Swans' strong defensive record, with no side in the Championship conceding as few as the 23 goals that they have shipped in 32 games this season.
Cooper will be desperate to pick up another win on Saturday, as that would set them up to cement themselves in the top two with a win on Tuesday night from one of their games in hand.
That will be no easy task though, as they welcome a Middlesbrough side who have their sights set on a playoff place.
Warnock's side got an important victory last time out, when they travelled to 20th-placed Coventry City on Tuesday night.
They suffered a poor start when Anfernee Dijksteel turned the ball into his own net after 10 minutes, before Grant Hall levelled the game and George Saville came off the bench to score a late winner to seal all three points for his side.
The side from the North-East now sit five points behind sixth-placed AFC Bournemouth, meaning they need to build a strong run of results in order to close that gap between themselves and the playoffs.
Warnock will be looking to celebrate his recent contract extension with what would be an impressive victory on Saturday, when his side travel to Wales.
Swansea City Championship form: WWLWLW
Swansea City form (all competitions): LWLWLW
Middlesbrough Championship form: LWWLDW
However, they recently welcomed Kyle Naughton back to action, and he went on to win the crucial penalty for the winning goal against Stoke City.
Yannick Bolasie returned to the Middlesbrough squad last time out, when he came off the bench in the win over Coventry after being out with an injury.
They will be without striker Ashley Fletcher however, as Warnock aims to get the attacker back for their game against Stoke City next week after picking up an injury against Bristol City.
Swansea City possible starting lineup:
Woodman; Naughton, Cabango, Guehi; Roberts, Fulton, Grimes, Hourihane, Bidwell; Lowe, Ayew
Middlesbrough possible starting lineup:
Bettinelli; Fry, Hall, Dijksteel; Spence, Tavernier, McNair, Howson, Bola; Kebano, Watmore
We say: Swansea City 2-1 Middlesbrough
While Warnock does know how to coach a resilient defensive outfit, we see his Middlesbrough side being outclassed by the Swans on Saturday.
The return of Hourihane and Bidwell should strengthen Cooper's side, inspiring them to another crucial victory as they look to break into the automatic promotion spots.