Wicket keeper Rhys Davies (50) and Clint McCabe's (75) 101-run partnership seemed to have set skipper Carlos Nunes' men on their way to atleast a winning draw as they 125-2 chasing Hornsey's 283-9dec.
But Twickenham, in the face of 4-74 from ex-Middlesex star Paul Weekes, lost their last eight wickets for 84-runs as they were dismissed for 209.
With league leaders Teddington losing by four wickets at North Middlesex, victory would have put Twickenham four points behind their derby rivals.
Earlier, Rashid Mullahzadah's 5-66 had seemingly rescued Twickenham from chasing a total close to 300 as Hornsey reached 170-3 thanks to 96 from skipper Michael Phillipson.
Mullahzadah's heroics saw Hornsey collapse to 189-8 before 66 from number nine Ed Wharton, who had not scored when he survived a big shout for caught behind, saw the tail wag to the tune of 94 runs.
Teddington were dismissed for 99 runs on Saturday as they were taunted by North Middlesex seamer Ed Bird's 8-44 and Richmond slipped into relegation trouble with a seven wicket defeat to Hampstead at Old Deer Park.
The hosts, who travel to Teddington on Saturday, struggled to 140 all out before Hampstead cruised to victory.
Fixtures: North Middlesex v Twickenham, Teddington v Richmond