The visitors led through Andy Forbes and a Craig McAllister penalty before Dave Tarpey threw Farnborough a 71st-minute lifeline.
The last 20 minutes were not quite as comfortable as the new Spitfires manager might have liked, but it was nothing he hadn’t expected.
“We tended to make a bit of hard work of it having taken an early lead, but their goalkeeper made a good save from Tom Jordan and we had a good penalty appeal turned down for handball,” said Hill.
“I’m not saying it was a great performance, but I was pleased with the last ten minutes or so when it got tough.
“The lads showed good character in respect of hanging on in there.
“We defended well and it was pleasing to see the lads looking organised and keeping to the shape I’d asked them to play in.”
It took the Spitfires just six minutes to get their noses in front when Andy Forbes headed home a Jai Reason free kick.
That lead was doubled just after the hour when Damian Scannell – recalled to the starting line-up following the surprise departure of skipper Daryl McMahon to Dover – was brought down and McAllister converted the penalty.
Dave Tarpey halved the arrears with a cross-shot in off the post following good work by Nic Ciardini, but Eastleigh held on to take their place in the third qualifying round where they will travel to Gloucester City on the weekend of October 6/7.
According to Winchester City, Eastleigh have lodged seven days’ notice of approach for former Swindon and Tottenham midfielder/defender Leigh Mills who has left the Denplan City Ground.
Hill said he knew nothing of the approach but confirmed he had been “aware of Leigh Mills for a number of years.”
* Ex-Saint Callum McNish, who finished a month’s loan at Eastleigh last week, has signed for BSB South rivals Basingstoke from Braintree.
The 20-year-old instantly made his mark, scoring in stoppage time to seal Stoke’s 3-1 FA Cup victory over Weymouth at the Camrose.