The redoubtable wicketkeeper and young all-rounder shared an unbroken sixth-wicket stand of 63 to see the hosts to a winning target of 193 in the dying embers of a fluctuating day.
Ollie Robinson (4-50) including a spell of 3-3 in 10 balls had threatened to be the scourge of Middlesex once more, but they were left to rue dropping catches at slip to reprieve both Andersson (27 not out) and Simpson (32 not out) in the dramatic finale.
Jason Gillespie, Sussex head coach, said: “It was a good game of cricket – unfortunately we are on the wrong side of the result. I thought the lads put in a shift with the ball both innings. With the bat, first innings we got a number of starts and put a number of partnerships together, but no really big ones.
“The last three years we are a county with among the most 50s with the bat, but one of the lowest conversions into 100s. There were a couple of moments where we probably let ourselves down. Just after lunch on day two we had Middlesex in a little bit of strife having bowled well that morning but let them get away from us a little bit where had we bowled with a little more discipline we would have had a bigger lead.
“Then yesterday afternoon we were in a spot of bother having lost a few quick wickets and that stops you driving the game forward and dictating terms. When you have those opportunities in a game you need to capitalise.”
Earlier in the day, Middlesex stalwart Tim Murtagh (5-34) recorded his 37th 5-wicket haul in first-class cricket as Sussex were hustled out for 102 in their second innings.
Andersson took 3-21 in the collapse, while Simpson held four catches behind the stumps.
Sussex began the day 33-4 in their second innings, a lead of 123, and not out batsmen skipper Ben Brown and Delray Rawlins rapidly increased the advantage with a flurry of early boundaries.
But Murtagh ended the revelry with one which stopped in the pitch causing Rawlins to lob the ball back to the bowler off the leading edge.
Murtagh completed his latest five-wicket haul soon afterwards when Tom Haines edged to Simpson and when Robinson was spectacularly caught by the wicketkeeper down the leg-side, Sussex were 74-7.
It might have been worse had a huge shout for LBW against Jack Carson been upheld when the spinner had made just two, an inside edge presumably saving him from being stone dead.
Brown’s nine fours ensured the visitors scraped their way past 100, but he was one of Andersson’s three victims while Miguel Cummins took 2-45.
Middlesex, needing 193 for victory had two overs to survive before lunch and knocked 17 off that total thanks to a fluent start from Max Holden.
The left-hander and fellow opener Sam Robson continued to accumulate after the break in contrasting fashions.
Holden played nicely, striking five fours, while Robson hung in in somewhat scratchy fashion against probing bowling from Robinson and Mitch Claydon.
The partnership had realised 48 when a game within a game began with Robinson the central character.
The seamer, whose career-best 8-34 Came against Middlesex last season struck Robson near the boot with the former England man playing no shot.
The appeal was huge, but not answered in the affirmative, following which Robinson appeared to make a TV replay gesture. That provoked a meeting between the umpires and skipper Brown and Robinson were spoken too.
A no-ball, one of several from Robinson, only served to get him ticking all the more, but his response was magnificent.
The third ball of his next over lifted on Holden, who appeared to glove it behind. The former England U-19 captain left shaking his head, whether in disappointment at his shot or the verdict was unclear.
One ball later in-form Nick Gubbins joined him after also nicking through to Brown and although Stevie Eskinazi kept out the hat-trick ball Robinson produced a brute of a delivery in the next over which Robson could only fend to Tom Clark at third slip.
Robbie White joined Eskinazi and the two rebuilt either side of a tea interval brought forward by a rain delay, adding 73 before part-time medium pacer Haines made what looked a vital incision trapping White LBW for 35 with 63 still needed.
Before that wicket Robinson had looked weary, but revitalised he trapped Eskinazi (26) to raise Sussex’s hopes once more. It was the 44th wicket claimed by Robinson against the Seaxes in their last five meetings.
It could have proved decisive if Phil Salt had held on to a slip catch offered by Andersson when he’d made just nine, while Finch grassed another nick from Simpson with 25 needed, but Middlesex made it home.