Pakistan, set a revised 162-run target in 38 overs under the Duckworth-Lewis formula, successfully achieved that in 34.1 overs on the back of an unbeaten 78 by Asad Shafiq and Mohammad Hafeez (49).
The victory gave the 1992 winners eight points from five matches and a place in the last eight, sending Zimbabwe packing after the first round.
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New Zealand, Sri Lanka and defending champions Australia make up the other qualifiers from Group A.
The final group standings will depend on the results of the remaining matches in the first phase. The top four teams will also qualify from Group B.
Pakistan's target looked tricky as Zimbabwe bowled a tight line and length. Left-arm spinner Raymond Price (2-21), opening the attack, dismissed opener Ahmed Shahzad (eight) in his second over.
Price and Utseya then bowled two maiden overs before Hafeez and Shafiq - playing his first World Cup match in place of injured Umar Akmal - opened up with some attractive drives.
Hafeez fell one short of his half-century, caught off Utseya after hitting six boundaries during his 65-ball knock.
Shahid Afridi promoted himself to number four but managed just three runs. Shafiq and Younis Khan (13 not out) saw Pakistan home. Shafiq hit seven boundaries during his sedate 97-ball knock.
Earlier, seamers Umar Gul (3-36), Abdul Razzaq (1-24) and Wahab Riaz (1-21) removed Zimbabwe's top four for just 43 after Elton Chigumbura won the toss and chose to bat in overcast conditions.
Spinners Afridi and Hafeez, with a wicket each, further derailed Zimbabwe, who at the first rain break were 96-5.
Left-hander Craig Ervine fought a lone battle, hitting a patient 52 off 82 balls with five boundaries for his third ODI fifty, before Hafeez bowled him in his first over.
Chigumbura (32 not out) and Prosper Utseya (18) added 48 for the seventh wicket.
Zimbabwe made a disastrous start, losing in-form Brendan Taylor off the fifth ball of the innings, caught off Razzaq by wicket-keeper Kamran Akmal.
Gul made it 5-2 when he dismissed Regis Chakabva off the last ball of the second over, before removing Vusimuzi Sibanda (five) shortly afterwards.
Ervine and Tatenda Taibu added 30 for the fourth wicket before Wahab Riaz, in for Shoaib Akhtar, had Taibu caught off a miscued drive at mid-off by Afridi for 19.
Zimbabwe play their last match against Kenya on March 20.
B Taylor c Kamran b Razzaq 4
R Chakabva lbw b Gul 0
T Taibu c Afridi b Riaz 19
V Sibanda c Misbah b Gul 5
C Ervine b Hafeez 52
G Lamb c and b Afridi 16
E Chigumbura not out 32
P Utseya c Kamran b Gul 18
Extras: (lb2, nb1, w2) 5
Total: (for seven wkts; 39.4 overs) 151
Fall of wickets: 1-5 (Taylor), 2-5 (Chakabva), 3-13 (Sibanda), 4-43 (Taibu), 5-84 (Lamb), 6-103 (Ervine), 7-151 (Utseya)
Bowling: Razzaq 7-1-24-1 (w1), Gul 7.4-1-36-3 (nb1), Riaz 6-0-21-1, Afridi 8-0-33-1, Rehman 8-1-24-0 (w1), Hafeez 3-0-11-1
Pakistan (target 162 in 38 overs):
Mohammad Hafeez c Price b Utseya 49
Ahmed Shahzad st Taibu b Price 8
Asad Shafiq not out 78
Shahid Afridi b Price 3
Younis Khan not out 13
Extras: (w13) 13
Total: (for three wkts; 34.1 overs) 164
Fall of wickets: 1-17 (Shahzad), 2-99 (Hafeez), 3-110 (Afridi)
Bowling: Masakadza 6-1-44-0 (w9), Price 8-1-21-2 (w3), Utseya 7-1-24-1, Lamb 7-0-44-0, Cremer 6.1-0-31-0 (w1)
Result: Pakistan won by seven wickets (via D/L method)