Finch batted for more than three hours to hit 12 fours in a top score of 69 out of the visitors’ total of 293 all out.
Delray Rawlins weighed in down the order with a run-a-ball 46, but Sussex’s day was marred by a back injury to opener Tom Haines, who returned late in the innings with a runner, but looked in great discomfort.
For Middlesex, there were four wickets for all-rounder Martin Andersson (4-77) and Blake Cullen, making his debut for the county, aged 18, returned figures of 2-51.
Jason Gillespie, Sussex head coach, said: “We’re happy with our score and I’m pleased we’ve got a few more partnerships together than in the last couple of games, so that’s a progression. But again, we are not ticking off those really big partnerships and going on to get those 100-plus stands or turn those good starts into three figures.
“We had Middlesex where we wanted them, and we just need to be ruthless in those situations. It would have been nice to get over 300 and go on to 350-plus, but we’ll certainly take what we’ve got because we know there’s enough in the surface to create problems.
“I thought Harry played really well today, but again he did all the hard work getting to 60 and there was an opportunity for a big hundred there. That is something we need to be better at.”
The morning session was a story of a missed opportunity for England hopeful Phil Salt. The Welsh-born opener battled through testing opening overs from Tim Murtagh and Miguel Cummins, surviving a scare when he lifted the former just over the head of the latter at mid-on. It was the cue for Salt to push on and he produced a gorgeous shot to loft left-arm spinner Walallawita back over his head for a straight six.
Another maximum followed when he pulled a short one from Cullen over the square leg fence, but his good early work was undone when he attempted to hook Andersson’s first ball bouncer and only succeeded in top edging it to Cullen at fine leg to depart for 42. It proved to be Middlesex’s only success pre-lunch as fellow opener Haines steered the visitors to 80-1 at the interval.
However, Haines hadn’t added to his tally after the resumption when he propped forward to play a defensive shot against Walallawita only to suffer a back spasm. He was briefly treated on the field before heading back to the dressing-room for intensive physio. His replacement Tom Clark struck two sweet boundaries before becoming the first first-class victim for Cullen, edging a ball which left him off the pitch to a diving John Simpson behind the stumps.
That brought Sussex skipper Ben Brown to the crease, another man who historically has flourished against Middlesex. He and Finch played nicely through much of the afternoon, each producing some sweet cover drives. They both survived scares, Brown almost turning a ball from Walallawita into the hands of short leg, while Finch drove loosely at Murtagh, the ball flying wide of the grasping hands of Sam Robson at slip.
It was something of a surprise when Brown (26) fell to the last ball before tea, mistiming a short ball from Murtagh (2-41) and spooning the ball gently into the grateful hands of Robbie White at mid-wicket. Finch found fluency after tea, plundering three fours from one Cummins over as he and new batsman Aaron Thomason raised the 200.
However, it would all end in disappointing fashion for Finch (69) as he fenced at a straight one from Cullen, giving Simpson the simplest of catches. Aaron Thomason meanwhile had taken 30 balls to get off the mark, but then only made 10 before Andersson pinned him LBW.
Delray Rawlins and Ollie Robinson stabilised matters again with a stand of 35 before Cummins returned to trap the latter LBW to one which looked a little high. If there was doubt about that Jack Carson was stone dead to the next delivery, but Mitchell Claydon survived a lifting hat-trick ball.
Rawlins cut loose with an enormous six over the old pavilion off Walallawita and three successive fours from the returning Andersson. The young all-rounder gained revenge when Rawlins holed out to Nick Gubbins on the fence, but his swashbuckling effort had edged Sussex ahead of the game once more.
The limping Haines returned with a runner, but Andersson had Claydon taken at slip before Murtagh scattered Will Sheffield stumps to end a fascinating day.