Marcus Harris made an unbeaten 100, reaching three figures from the day’s penultimate delivery when he pulled a short ball from left-arm spinner Delray Rawlins for four, while right-hander Burns finished the day on 88.
Together the pair put together a partnership of 203 from just 49 overs, and with two days left to play their union leaves Australia A in charge after it had earlier dismissed Sussex for 263.
Australia A’s remaining batsmen will hope for similarly fruitful opportunities as the bid to seal slots in the senior side intensifies ahead of the first of five Tests against England that begins at Edgbaston, Birmingham on 1 August.
Harris’s twelfth first-class hundred arrived in 142 balls and included 16 fours and he was especially strong square of the wicket. Burns, meanwhile, has so far struck 13 fours and played some very pleasing drives both along the ground and, when the spinners appeared, over the infield.
Sussex was without several key members of its frontline attack, including fast bowler Ollie Robinson, named for next week’s England Lions fixture at Canterbury, England all-rounder Chris Jordan and the South African David Wiese.
And in their absence, the home side’s bowlers found it impossible to apply a brake to Australia A’s scoring as captain Luke Wells used six bowlers trying to stem the bleeding.
Wicket-taking was also a tough assignment in the first part of the day as Sussex progressed from a precarious overnight 5-118 to eventually total 263.
Fast bowler James Pattinson (4-60) finished as Australia A’s most successful bowler but it was left-arm spinner Jon Holland (3-49) who proved captain Tim Paine’s trump card on day two, offering accuracy and patience, key commodities to help winkle out a stubborn lower order.
Seamer Jackson Bird (1-44) was Australia A’s other wicket-taker, and although Josh Hazlewood (0-46) went wicketless, all the attack will have benefited from the chance to perform in testing conditions.
The match against Sussex runs until Wednesday 10 July, after which the Australia A squad will relocate to Canterbury for a four-day match against the England Lions starting on 14 July.
The tour will then conclude with another four-day game, this one involving the senior Australia side, in Southampton from 23 July, before the Ashes series gets underway on 1 August.
The Australia A squad has already played four one-day matches on the trip, against Northamptonshire, Derbyshire and Gloucestershire.