A 20th-minute Ryan Curran header looked set to give Harps victory before Josh Daniels' 84th-minute goal was followed by a Rory Patterson strike on 88.
But Houston powered in a magnificent header in the first minute of injury-time to level.
Nathan Boyle missed a glorious Derry chance with the last kick of the game.
Leaders and champions Dundalk are now seven points above the Candystripes after edging a 1-0 home win over Bray on Tuesday evening.
Cork City's 1-0 win in Dublin over Bohemians keeps them four points behind Dundalk while Shamrock Rovers move above Derry on goal difference into third spot after beating Sligo Rovers 3-0.
With former Brandywell favourite Barry Molloy in fine form for Harps, the visitors were good value for their 20th-minute lead as Curran nodded into the net after a David Scully shot had come back off the woodwork.
Derry did improve before half-time with Patterson wasting two good chances and impressive Harps keeper Ciaran Gallagher also denying Ronan Curtis.
As Harps retreated into a defensive shell in the second half, Derry dominated possession but they needed the introduction of youngster Daniels to finally breach their visitors.
Daniels, who lost five members of his family including his mother and sister in the Buncrana pier tragedy in March, notched his first senior goal for Derry on 84 after good work by the impressive Niclas Vemmelund and Boyle.
Youngster Daniels then set up Patterson to strike Derry ahead four minutes later with the shot appearing to take a slight deflection.
To their immense credit, Harps refused to give up and a clever Kevin McHugh ball released Michael Funston whose cross was brilliantly headed to the net by Houston from the edge of the penalty area in the first minute of injury-time.
Inevitably, there was still time for more drama as Boyle somehow managed to side-foot wide from three yards with the goal gaping after a Vemmelund cross.
Derry lost 2-1 against Harps in their Premier Division opener in early March.

"The joke has worn thin. I'm tired of having your opinions ascribed to me. Pls tweet under your own name. Thanks", the author had tweeted to the account.
The handle with more than 30,000 followers adopted a "faux Rushdie persona" and engaged in debates with right-wing accounts.
The owner immediately capitulated and changed his handle to @IndiaExplained.
"It gave me a chance to pay homage to Rushdie and other writers while being able to reflect and comment on things that matter to me", the owner of the account, Rohit Chopra told Indian news website Scroll.in.
But many Indians online have taken exception to Sir Salman's stance, accusing him of both lacking a sense of humour and clamping down on freedom of expression.
With its new name, the account has promised to provide "satirical and political commentary on India".
It is just one among a number of Indian parody Twitter accounts which have courted controversy with those tried to send up.
In one instance, the office of former Indian Prime Minister Manmohan Singh asked Twitter to block the account of @PM0India (which used a zero instead of a 'O') because of "objectionable content", which "could be mistaken as the official account of the PMO and have serious ramifications."

County Durham and Darlington Fire and Rescue crews were called to Beech Close, Brasside, on Monday evening. The street is in the Framwellgate Moor area of the city.
The man was pronounced dead at the scene.
A joint investigation between the fire service and Durham Police has been launched, a fire service spokesman said.

25 November 2016 Last updated at 07:57 GMT
It was given to the London museum in 1905 by a rich business man and the giant skeleton has been amazing visitors for many years as they enter the museum's main hall.
It isn't a real dinosaur but a life size plaster replica of the skeleton of a Diplodocus.
Now Dippy is on the move on a trip around the UK, watch this video to find out more.
Derry City dropped to fourth in the League of Ireland table after Sean Houston's injury-time goal earned Finn Harps a dramatic Brandywell draw.
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Salman Rushdie has been pilloried by Indian tweeters after confronting a parody account @RushdieExplains.
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A man has died in a house fire in County Durham.
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Dippy the Diplodocus is one of the Natural History Museum's most recognisable residents.