Coyne opened the scoring from the penalty spot in the 63rd minute after Eddie May had been brought down by Ray Sharp. He almost grabbed a second when he rattled the crossbar with a 78th-minute header.
Dunfermline brought on two substitutes - Alan Moore for John Fraser and Stewart Petrie for Sharp - and for a time it looked as if the changes might swing the game back in their favour.
But Coyne put the result beyond doubt five minutes from time when he cheekily backheeled home a Weir free-kick.
Tempers frayed at one point and there were six bookings - John Fraser, Andy Smith and Greg Shields for Dunfermline and Franz Resch, Steve McMillan and Greg Denham for Motherwell.
St Johnstone were thankful to an old campaigner as they marked their return to the top flight after a three-year absence with a 1-1 draw - but it was not a man drawn from within their own ranks.
The former Aberdeen defender Stewart McKimmie put through his own net after 61 minutes to deny Dundee United the opening-day victory their dominance deserved at McDiarmid Park. McKimmie was powerless as Atilla Sekeriloglu's powerful cross-shot diverted off him and past Sieb Dykstra to cancel out Kjell Olafsson's 11th-minute opener. It had looked like the Swede's early goal, volleyed in off a post, would be the first of many at that stage, but Saints hung on grimly before the unfortunate McKimmie ensured them a point.
The result was tough on Tommy McLean's side, who looked composed throughout but were unable to claim the goals that their possession warranted.
The lure of new summer signings and the visit of the Tennents Scottish Cup winners Kilmarnock attracted an above average crowd of over 14,000 to witness a frustrating goalless draw at Pittodrie, as Aberdeen failed to get the goal that their control of the game deserved.
Aberdeen's best chances fell to Dean Windass, who headed off the underside of the bar, and Billy Dodds, whose late header was denied by a superb diving save by Dragoje Lekovic six minutes from time.
Killie, who gave the former Scotland and Tranmere winger Pat Nevin a second-half debut, battled hard for their point but could have had all three had the French substitute Jerome Vareille succeeded in scoring in the 70th minute when, with his first touch, he thundered a 12-yard volley off the bar.Reuse content