ScotRail sent a rescue train and transferred passengers using step ladders.
They had been stuck for about three-and-a-half hours.
A tree branch is believed to have caught in the wires, with electrical equipment on the roof of one of the trains also reported to be damaged.
The incident at 8:20am is expected to close the lines between Glasgow, Gourock and Wemyss Bay for the rest of the day.
Later, a ScotRail train with 100 passengers on board was able to drive slowly through debris from another landslip on the Glasgow-Mallaig line at Glen Douglas beside Loch Long just before 2pm - the third weather damage on the line in five days.
The Jacobite steam train also struck a tree between Fort William and Mallaig.
The Met Office warned of further downpours until Sunday, with as much as 80mm (3in) of rain possible over parts of southern Scotland within as little as six hours.
Following the Bishopton incident, a train shuttle service was introduced between Glasgow Central and Paisley Gilmour Street, with replacement buses over the rest of the routes.
Hayden Sutherland, who had just boarded a Glasgow-bound train at Bishopton when the incident happened, praised how staff had handled it.
The digital consultant told The Scotsman: "The train stopped and the lights dimmed.
"Then the driver came on to explain a tree had taken out a power line, and he continued to keep us up to date.
"Staff were profusely apologetic, even though it wasn't their fault.
"They worked above and beyond the call of duty, and ensured everyone got off the train safely.
"Everyone was really, really helpful."
Mr Sutherland also tweeted his appreciation for ScotRail driver team manager Colin: "Huge thanks to the wet but smiling @ScotRail staff for delivering the 'rescue train'."
ScotRail tweeted: "The overhead lines are down between Langbank and Bishopton.
"The rescue train is now on-site and we're starting to transferring customers across.
"This service will then travel to Bishopton where customers can continue their journey by replacement bus."
On the landslide, Network Rail Scotland tweeted: "Due to a further landslip between Arrochar and Glen Douglas, the southern part of the West Highland Line is currently closed.
"We’ve dispatched specialists to assess the situation and confirm what work is required to reopen the line."
ScotRail said the driver of a northbound train had spotted landslide debris and drove through it at 1mph so passengers could reach Arrochar station rather than have to be taken off the train.
Meantime, ScotRail said Jacobite operator West Coast Railways had reported that its 10:15am service had "struck a tree".
"The driver is worried that he may cause more damage to the infrastructure by continuing along the line."
Network Rail Scotland said the incident happened at about 12:15pm near Lochailort and the tree was cleared and line re-opened at 1:50pm.
Further south on the route from Glasgow, the section between Ardlui at the northern end of Loch Lomond and Crianlarich, will now remain shut until Thursday, 22 August after heavy rain damage on Sunday.
The Crianlarich to Oban line is due to re-open on Monday following repairs to separate damage at Tyndrum.
On the roads, surface water caused traffic queues on major routes including around the Queensferry Crossing.
Concrete deterioration of the River Almond bridge, thought to be weather-related, forced the closure of one lane of the M9 in Edinburgh.
A fallen tree caused problems for drivers on the A92 in Glenrothes, while strong winds posed a hazard on the A9 Dornoch Bridge.
CalMac cancelled its Gourock-Dunoon ferries and also sailings between Mallaig and Armadale on Skye because of strong winds, with several other west coast routes also disrupted.
The Met Office’s yellow - be aware - severe weather warning for heavy rain remains in force for the whole of Scotland apart from the west Highland and Islands until 6am on Sunday.
A spokeswoman said: “Heavy showers and thunderstorms are expected to develop through Saturday, lasting into the first part of Sunday.
“Whilst many areas will miss the worst of these thundery showers, a few spots could see 15-25mm within an hour and 30-40mm within 3 hours.
“It is possible some of these may become slow-moving and prolonged, most likely across parts of southern Scotland, leading to the potential for 60-80mm (up to 3in) of rain in 6-12 hours.”