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Adnan Akmal scored 74 not out as the hosts made 192, losing 12 wickets on day one of a two-day match as both teams agreed to spend a day batting.
Finn took 4-16 but will battle with Mark Wood for the third seamer's place.
Meanwhile, leg-spinner Adil Rashid, pushing for a Test debut, took 3-53.
Finn, who took eight Australian wickets to win the third Ashes Test in July, and Wood played alongside Stuart Broad in the last two Tests against Australia after James Anderson was injured.
But Finn expects Wood, Broad and the fit-again Anderson to be selected for the first Test, which begins on 13 October in Abu Dhabi, after the other three all played in the opening warm-up game earlier this week.
"I'd say I've staked a claim but I'd still be surprised if I played the first Test," said the Middlesex bowler, who took the new ball with fellow paceman Liam Plunkett.
"I think the guys bowled really well the other day, and it wouldn't surprise me if the team that was picked then is the team for the first Test.
"At the moment, I'm back-up to those guys but nothing is set in stone."
Finn's match-winning Ashes performance at Edgbaston came after he doubted if he would play Test cricket again during his two-year absence from the longest format.
On that occasion, Wood was rested and Finn believes it will be a straight pick between him and the Durham seamer.
"There will be disappointed people, regardless of who gets picked," added Finn. "I don't think the same attack is going to play all the way through the series.
"So if you don't play the first Test, it's not the end of the world - there are two others in very quick succession."
Attention will turn to England's batting on Friday, with Moeen Ali and Alex Hales both auditioning for the role as captain Alastair Cook's opening partner.
England pace bowler Steven Finn pressed his claims for a place in next week's first Test against Pakistan after taking four wickets in their final warm-up against Pakistan A in Sharjah.