Luke McCowan gave the home support something to cheer after 26 minutes when he galloped clear of the Falkirk defence and kept his composure to finish.
Declan McManus equalised with a deflected shot which deceived goalkeeper Ross Doohan just before half-time. Falkirk were reduced to ten men shortly after the break when Mark Durnan and Doohan clashed, the goalkeeper being yellow-carded. Andy Geggan slammed the ball home to put Ayr back in front before Ian McShane saw a penalty saved by Doohan as Ayr held on.
Stranraer kept in touch by thumping troubled Berwick Rangers 6-0. Cameron Elliott put them in front midway through the first half and Jordan Allan got on the scoresheet seven minutes from the break. There was no way back for Berwick when Conor McManus and Mark Stewart (penalty) both added goals before the break. James Hilton made it 5-0 midway through the second half before Stewart added his second penalty.
In Group D, Inverness Caledonian Thistle kept in touch with the leaders with a convincing 4-1 win over Raith Rovers. Jordan White headed them in front from a James Keatings corner and, after Lewis Allan equalised, Coll Donaldson, Aaron Doran and substitute Nikolay Todorov completed the scoring in the final moments with a header.
In Group G, East Kilbride drew 0-0 with an unconvincing St Mirren and took the bonus point after winning the penalty shoot-out 6-5. Goalkeeper Alan Martin was the star man, producing a brilliant double save from Danny Mullen late in the first half and St Mirren could not find a way past him.
In Group F, Hamilton Academical and Partick Thistle drew 2-2, with Hamilton taking the penalty shoot-out 6-5. Ryan Williamson scored early for Partick but a brilliant double from Ross Cunningham turned the tables. Shea Gordon equalised before Aaron McGowan skied a late penalty for Hamilton, sending it to a shoot-out.
Dumbarton fell 4-1 at home to Queen of the South in Group E, all of the goals coming in the first quarter. Dumbarton took an early lead through PJ Crossan, but it was short-lived as Queen of the South hit back with two goals in as many minutes. Michael Paton equalised and before Dumbarton could regroup, Gary Oliver put the Dumfries side in front. Stephen Dobbie then netted the third and Connor Murray made it 4-1.
In Group C, Elgin City eased to a 3-0 win over Stirling Albion. In-form Shane Sutherland contributed a double and Kane Hester was also on target.
Forfar Athletic kept up their impressive start in Group B with a 4-1 win at Montrose. Josh Skelly gave Montrose an early lead from the penalty spot but Dale Hilson equalised with another penalty after 25 minutes and then put Forfar in front with his second penalty. Michael Travis netted a second-half double to make it 4-1.
In Group A, East Fife beat Stenhousemuir 2-0 with Scott Agnew grabbing both goals, his second from the penalty spot.