The Wales international, whose father David played for the Tractor Boys between 1997 and 2001, bagged a first-half double, while Djed Spence and Dejan Kulusevski were on target later on to seal all three points at Portman Road.
Johnson was in the right place at the right time twice in the space of eight minutes to convert a pair of Son Heung-min assists, though Omari Hutchinson halved the deficit before the break with an impressive first-time strike for the hosts.
The result remained in the balance until the closing stages, when a deflected Spence effort and Kulusevski's fine individual goal ultimately condemned Ipswich to a seventh defeat in eight home league matches.
Kieran McKenna's side remain 18th in the league table, but they are now five points from safety following Wolves' surprise win at high-flying Bournemouth.
Spurs, meanwhile, stay 12th and are six points adrift of the top 10.
Around this time 25 years ago, David Johnson's goals were instrumental towards Ipswich sealing their return to the Premier League. But a quarter of a century later, his son’s goals have dented their hopes of remaining in the top flight.
Johnson's first Premier League brace for Tottenham took his tally for the season to nine, his highest during a single campaign.
Both were set up by Son, who has been directly involved in eight goals (two goals, six assists) in his last eight away league games.