Lanning made 69 off 49 balls as the Capitals launched their pursuit of UP's 7-166, but was bowled by compatriot Grace Harris with 48 needed off 32 balls.
With the rate beginning to climb Sutherland found herself needing to hit ten runs off the last five balls, bowled by Ashes-winning teammate Tahlia McGrath.
She swiped the first to wide mid-wicket where Alana King and Harris ran to intercept, but somehow got in each other's way allowing a four. The next ball was also cracked for four, through extra cover.
Sutherland then grabbed a single.
With one to win, two to bowl, in came the fielders. Marizanne Kapp, a South African interloper into this Aussie showdown, stabbed the ball to mid-on and took off. Harris gathered, threw to McGrath - who failed to collect the ball and thus missed the run-out.
Capitals were home, by a ball and seven wickets to spare.
"We like a close finish, happy to be on the right side of it," said Lanning, whose team scrambled a two-wicket win off the last ball in their opening game.
Capitals lost their second game and Lanning added: "There's always areas to improve, but I feel this was our best performance of the tournament so far.
Of her own innings she said: "I felt I was positive from the first ball. Got my footwork going."
Having preceded her unbeaten 41 (off 35 balls) by taking 2-26, ending Warriorz' 66-run start partnership by dismissing both openers, Sutherland was player of the match.
"I just try to find out what's working as early as possible," she said. "It's tough for bowlers at times. Sometimes it's your day, sometimes it's not."
Sutherland was helped by being gifted a life on 23 by Sophie Ecclestone, the Englishwoman continuing to suffer the butterfingers affliction prevalent in the Ashes, spilling a simple chance at mid-on. The bowler was McGrath, suddenly less thrilled at English failings.
King, on as a substitute fielder, also dropped Kapp late on, though that was a harder chance.
The other Australian in the game was Jess Jonassen, who bowled tidily to take 1-21 off four overs, the most economical return of the match, and picked up the wicket of McGrath (1), one of two stumpings by Scot Sarah Bryce.
Capitals moved to second in the five-team table, level on points with Royal Challengers Bengaluru, who include Ellyse Perry, Georgia Wareham, Heather Graham and Kim Garth. Warriorz are bottom, and winless, but with a game in hand on all except RCB.
