Injuries have led head coach, former Gala ace Gregor Townsend, to reshuffle the squad, with another ex-Mansfield Park star, Stuart Hogg, still absent because of the shoulder injury he suffered against Ireland.
Sean Maitland is also out of contention after a foot injury – which means Edinburgh winger Graham will line up alongside Tommy Seymour and Blair Kinghorn in the back three.
The 21-year-old played for the final 25 minutes or so last time out against France, where the commentators made much of his speed and ability.
However, Graham only had a couple of opportunities to show his pace, as the Scots went down 27-10, so a first full start for his country may yield better fortunes.
Tommy Seymour and Glasgow Warriors colleague Jonny Gray will make their 50th appearance in a Scotland side , while the revamping among the half-backs will hopefuly boost the hosts’ efforts to secure the Doddie Weir Cup.
Racing 92 stand-off Finn Russell returns from a head injury to start alongside Glasgow Warriors scrum-half Ali Price. Russell’s return sees Pete Horne move one place outfield to inside-centre – in place of Sam Johnson – to partner fellow Glasgow Warriors centre Nick Grigg.
The inclusion of Price means Greig Laidlaw, from Jedburgh, moves to the bench, with Edinburgh hooker and club captain Stuart McInally taking on the national team responsibility for the third time.
The other players with Border links who have figured recently in the squad – Gary Graham and Jamie Bhatti – are out of contention for a place this time around.
The only change to the pack is at tighthead, where Edinburgh prop Willem Nel returns from a calf injury to form an all-Edinburgh front-row with McInally and loosehead Allan Dell.
They pack down in front of an unchanged back-five where lock Grant Gilchrist partners Gray in the second-row and Magnus Bradbury, Jamie Ritchie and Josh Strauss return to the back-row.
There are three further changes to the bench, which could see recent squad additions Hamish Watson (back-row), Gordon Reid (loosehead prop) and Byron McGuigan (back three) feature for the first time this campaign.
Gregor Townsend explained: “While there have been some strong moments in our games this campaign, we have yet to deliver that quality and focus for 80 minutes and play to our potential.
“We have had a couple of weeks since our defeat in Paris to look at ways we can improve our performances, through what we do in training, our tactical approach and through selection.
“First and foremost, we select a team we believe gives us the best chance of winning at the weekend, while we also have an opportunity to reward players who have been in very good form and have been knocking at the door of the team in recent weeks – Darcy Graham and Ali Price especially.
“Greig Laidlaw, having started in our last six games is a key member of our squad, although this week we have decided to utilise his leadership and experience off the bench.
“He is driven by a desire to win for his country and he should have an important role to play in the latter stages of the match at the weekend.”
Wales – the world’s third-ranked side – arrive in Edinburgh as the only unbeaten team remaining in the championship and have won 10 in their last 11 Tests against their hosts, with Scotland’s success coming in their last meeting at BT Murrayfield two years ago.
Townsend added: “Wales have a very good record against us and are in excellent form. It’s a similar challenge to the one we faced last year against England, who had also been on a very good run of results in their preceding games.
“We rose to the challenge that day and found a way to win and I firmly believe we can reach that level of performance once again on Saturday.”
Scotland team to play Wales at BT Murrayfield:
15 Blair Kinghorn (Edinburgh), 14 Tommy Seymour (Glasgow Warriors), 13 Nick Grigg (Glasgow Warriors), 12 Pete Horne (Glasgow Warriors), 11 Darcy Graham (Edinburgh), 10 Finn Russell (Racing 92, vice-captain), 9 Ali Price (Glasgow Warriors); 1 Allan Dell (Edinburgh), 2 Stuart McInally (Edinburgh, captain), 3 Willem Nel (Edinburgh), 4 Grant Gilchrist (Edinburgh, vice-captain), 5 Jonny Gray (Glasgow Warriors), 6 Magnus Bradbury (Edinburgh), 7 Jamie Ritchie (Edinburgh), 8 Josh Strauss (Sale Sharks) 20 caps. Subs: 16 Fraser Brown (Glasgow Warriors), 17 Gordon Reid (London Irish), 18 Simon Berghan (Edinburgh), 19 Ben Toolis (Edinburgh), 20 Hamish Watson (Edinburgh), 21 Greig Laidlaw (Clermont Auvergne), 22 Adam Hastings (Glasgow Warriors), 23 Byron McGuigan (Sale Sharks).