The Scotland international, 22, has joined from the Scottish champions for a fee of £25million.
The left-back, who will wear the number three shirt, passed a medical in London on Thursday and could make his debut against Newcastle on Sunday.
Manager Unai Emery said: "We're delighted Kieran is joining us. He's a very talented player who will continue to improve.
"He increases our options defensively and I look forward to him joining our group."
Meanwhile the Gunners are still expected to confirm the signing of David Luiz from Chelsea following a falling out in west London.
The 32-year-old was at Chelsea's Cobham training ground on Wednesday, but it is understood he did not take part in the main training session with Frank Lampard's men.
Luiz signed a new two-year contract in May, with Chelsea straying from their standard practice of handing one-year extensions to players aged 30 or older.
Arsenal boss Unai Emery has worked with Luiz before and was desperate to add a new centre back before Thursday's deadline.
In terms of outgoings Alex Iwobi appears destined for the exit.
The club record arrival of Nicolas Pepe from Lille appears to have closed off any chance of regular first team football at the Emirates.
And Everton have pounced according to the BBC.
They report that the Toffees submitted a deal sheet and that the player was undergoing a medical in London ahead of signing a five-year deal.
In other exits Leeds won the race for forward Eddie Nketiah.
The 20-year-old joins the Championship club on loan for the season.
Bristol City were interested in Nketiah and had been front runners last month before Unai Emery felt the striker would stay at the Emirates Stadium this term.
But the Gunners have now decided to let him leave, despite impressive pre-season performances.
Leeds director of football Victor Orta has a good relationship with Emery and it is understood he made a personal pitch to Arsenal for Nketiah on Tuesday.
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