Motherwell striker Lukas Jutkiewicz had two early attempts before Hearts almost took advantage of some nervous goalkeeping by Michael Fraser.
Pavels Mihadjuks was withdrawn as keeper Michael Fraser was introduced from the Inverness bench and Arfield was successful from the spot.
Ermir Strati put the Albanians in front in the 66th minute when his 20-yard strike beat Michael Fraser and went in off the post.
Brown was also delighted for goalkeeper Michael Fraser, whose mistakes against Dundee United helped the Tayside club to a win at Fir Park last week.
John Sutton passed up several chances on either side of the interval to restore parity and Michael Fraser kept out a Jack Compton strike for Falkirk.
And Jim O'Brien spurned a chance to seal the points for the Steelmen late on, but his speculative effort from Michael Fraser's sliced clearance drifted wide.
Jeffrey De Visscher then found himself in space on the edge of the Inverness penalty area and shot towards Michael Fraser's right-hand post but the goalkeeper parried wide.
Hamilton started to take the initiative and Mark McLaughlin was unlucky to see his header go over Michael Fraser's bar when he was first to a Stevenson corner.
Inverness defender Lionel Djebi-Zadi's failure to control the ball inside his penalty area, the ball broke kindly to the Welshman who coolly side-footed past Michael Fraser from 10 yards.
And, although Motherwell missed a series of half chances, it was the visitors who grabbed a 64th-minute consolation when Roshi ran on to a simple through ball and slotted past Michael Fraser.
"The fundamental fact is cyber criminals are highly organized with sophisticated corporate structures and business chains, " said Michael Fraser, director of the Communications Law Centre at the University of Technology Sydney in Australia.
Esson's introduction had been one of three changes made by Caley Thistle manager Craig Brewster, goalkeeper Michael Fraser and strikers Roy McBain and Andy Barrowman paying the price of last weekend's 5-0 hammering by Rangers.
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