The Durham batsman followed up his brutal 137-ball 148 in the thrashing of Fiji on Wednesday with another high-quality innings from number three, hitting an unbeaten 106 from 104 balls.
Lawrence and Holden made good on Brad Taylor’s decision to make first use, sharing a 91-run opening stand.
Lawrence added 59 to his scores of 174 and 55 in the first two matches to become the tournament’s leading run-scorer, only to be knocked off top spot by Burnham.
He struck five fours and half-a-dozen maximums during his run-a-ball ton and contributed to half-century stands with Holden, who made 51, the recalled Aneurin Donald and Curran.
Curran gave the innings impetus by hitting five fours in a 16-ball 32 and then carried the momentum into his bowling, having Akshay Patel caught at mid-off in the fourth over.
Mahmood then took a wicket either side of lunch, becoming the the tournament’s leading wicket-taker in the process, to leave Zimbabwe’s chase in tatters at 14 for three.
Ives and Shaun Snyder offered brief resistance with a 35-run partnership, but after Ben Green removed the latter Callum Taylor took centre stage.
The right-arm seamer bowled with good pace to take 3-14 from four overs, including taking two wickets in as many balls, the first of which arrived courtesy of a brilliant diving catch at short cover from Brad Taylor.
He could not become the first England Under-19 international since Luke Wright in 2003 to take a hat-trick, though, delivering a leg-side wide.
Callum Taylor bowled Wesley Madhevere and, despite a battling 52-run eighth-wicket alliance between Ives and Brandon Mavuta, Mahmood and Mason Crane wrapped up the tail.
Mahmood denied Ives, who was trying his best alongside last man Rugare Mugarira, a ton by bowling him to return figures of 4-39 from 9.4 overs.