Twenty20 against Australia in Cardiff, started to choke off the supply of runs.
Burns fell when he hit a Rashid full-toss back to the bowler to end a first-wicket stand of 76.
Warner, unusually, was scoring at under a run-a-ball and went when, beaten in the flight by Rashid, the left-hander edged an intended drive to Woakes at short third man.
George Bailey and Smith both hit sixes off Rashid.
But Smith, one ball after clearing the rope, hit a Rashid full-toss to Ben Stokes at mid-wicket and was out for 44.
Rashid struck again when the 27-year-old Yorkshireman had Bailey lbw for 23 with a quicker ball that skidded on.
When Watson, responding to a poor call by Wade, was run out, Australia were 193 for six.
But Wade and Marsh added 93 runs in the last 10 overs, with their stand a seventh-wicket record for Australia against England in ODIs.
Jason Roy, driving the ball stylishly, and Alex Hales, put on 70 for England’s first wicket before Hales carelessly holed out off Marsh.
But South Africa-born Roy went on to complete a maiden fifty at this level, off 49 balls, including 10 fours.
Roy’s innings ended when, backing away, he sliced off-spinner Maxwell to point on 67.
Taylor made a near run-a-ball 49 before, advancing down the pitch to medium-pacer Watson — who played on this ground for Hampshire — he was bowled.
England then lost three wickets for no runs in four balls — including Morgan for 38 — with Nathan Coulter-Nile taking two in two, before Rashid prevented the hat-trick. The five-match series continues at Lord’s on Saturday (today).
Score Board
Australia won toss
Australia
J A Burns c & b Rashid 44
D A Warner c Woakes b Rashid 59
*S P D Smith c Stokes b Rashid 44
G J Bailey lbw b Rashid 23
G J Maxwell c Buttler b Wood 15
S R Watson run out 6
†M S Wade not out 71
M R Marsh not out 40
Extras (lb 1, w 2) 3
Total (6 wickets; 50 overs) 305
Did not bat: M A Starc, N M Coulter-Nile, P J Cummins
Fall: 1-76, 2-133, 3-164, 4-178, 5-192, 6-193
Bowling: Finn 7-0-41-0; Wood 10-0-72-1 (2w); Woakes 9-0-57-0; Rashid 10-0-59-4; Ali 8-0-37-0; Stokes 6-0-38-0
England
J J Roy c Warner b Maxwell 67
A D Hales c Watson b Marsh 22
J W A Taylor b Watson 49
*E J G Morgan c Wade b Watson 38
B A Stokes c Burns b Starc 13
†J C Buttler c Marsh b Coulter-Nile 4
M M Ali c Marsh b Cummins 17
C R Woakes c Wade b Coulter-Nile 0
A U Rashid c Wade b Starc 11
M A Wood not out 10
S T Finn b Cummins 10
Extras (lb 1, w 4) 5
Total (all out; 45.3 overs) 246
Fall: 1-70, 2-112, 3-152, 4-172, 5-194, 6-194, 7-194, 8-220, 9-232, 10-246
Bowling: Starc 10-0-55-2 (1w); Coulter-Nile 8-1-39-2 (1w); Cummins 9.3-1-48-2 (1w); Marsh 4-1-35-1 (1w); Watson 8-0-39-2; Maxwell 6-0-29-1
Result: Australia won by 59 runs
Series: Australia lead the 5-match series by 1-0
Man of the Match: M S Wade (Australia)
Umpires: M A Gough and J S Wilson (West Indies). TV umpire: H D P K Dharmasena (Sri Lanka). Match referee: J J Crowe (New Zealand)