The pair put on an unbroken stand of 105 to prevent a heavy defeat after Sussex had slumped to 123-4 following on.
Wright had scored 77 in the morning as Sussex were bowled out in their first innings for 222 just before lunch, a deficit of 277.
Middlesex opted to enforce the follow-on and it looked to be the right decision as Sussex were guilty of giving away their wickets cheaply once again.
But Joyce and Wright took Sussex to 228-4 at the close but they still trail by 49 and face a difficult task to save the game on the final day.
Wright and Will Beer had given Sussex a glimmer of hope in the morning with a partnership of 88 which occupied 40 overs.
The pair took the score past the 200 mark to earn a first batting point but the departure of Beer 20 minutes before lunch saw Sussex lose their final three wickets in quick succession.
Beer – who was dropped on 13 – faced 141 deliveries before being caught behind off the bowling of Neil Dexter for 25.
Wright obviously felt he had to grab the initiative following the loss of Beer but it backfired as he was out off the very next ball.
The England all-rounder gave a return catch to Ollie Rayner as he charged down the wicket at the former Sussex spinner having made 77.
Middlesex wrapped up the Sussex innings in the last over before lunch when Steve Magoffin was lbw to give Rayner his first five-wicket haul in the Championship since 2008.
Chris Rogers opted to enforce the follow-on during the lunch interval and Sussex’s hopes of an improved showing in their second innings took an immediate blow when they lost Luke Wells – who did not look pleased to be given lbw - in the first over.
Chris Nash and Mike Yardy put on 67 but their partnership came to an end in ridiculous circumstances as Nash was run out by a direct hit from Joe Denly trying to scamper a risky single.
Yardy then got in a horrible mess attempting to pull a short ball from Jamie Harris and skied a catch to Rogers just before having made 41.
That brought Matt Prior to the crease with a point to prove following his controversial dismissal on Thursday.
It proved to be another unhappy experience at Lord’s for Prior though – after making a pair of ducks in the Test against New Zealand – as he slapped a catch to mid-on off Harris when on 18.
A defeat inside three days looked a possibility at that point but Wright and Joyce began changing the complexion of the game.
The pair batted positively but needed a slice of luck too with Joyce dropped on 58 by Gareth Berg and Wright put down on 40 by Tim Murtagh.
Joyce will resume in the morning on 75 – his 50 coming off 86 balls – with Wright on 48.