Inter began life under their new coach with two league wins and had the chance to go eight points clear of title rivals Juventus following their defeat at Napoli, but the Nerazzurri had to settle for a point from an end-to-end encounter.
"There is some regret because it was a match that we had to win and wanted to win," Inzaghi told Sky Italia.
"We went ahead twice, then we missed three simple chances. Winning today would have been very important. The performance was there, all things considered, but there is some regret."
Lautaro Martinez ruthlessly finished a sweeping counter-attack with a volley to put the visitors back in front before half-time, but Tommaso Augello responded with a wonderful volleyed strike of his own early in the second half.
Both sides had chances to take all three points but neither found the decisive breakthrough, leaving Inter second with seven points and Sampdoria in 13th place with two points.
Inter scored seven goals during back-to-back wins over Genoa and Hellas Verona before the international break, but Inzaghi was left frustrated on his 200th Serie A game as a coach as Sampdoria earned a deserved point.
His side's task was made more complicated in the closing stages when Inter's fifth and final substitute Stefano Sensi went down injured and was unable to continue.
"Unfortunately we found ourselves down to 10 (men) for the final 20 minutes and maybe even risked losing," Inzaghi said.
"It leaves a bitter taste, but now we must get our players back because we have a lot of matches, including a very important one on Wednesday [at home to Real Madrid in the Champions League]."
The hosts, also under new management in Roberto D'Aversa, had a good early chance when Morten Thorsby headed over from six yards, but they went behind when Dimarco swept in an unstoppable free kick, which Sky Italia clocked at 65 mph.
Sampdoria equalised in the 32nd minute when a headed clearance from a corner fell to Yoshida, whose shot was deflected in.
But Inter retook the lead with a well-executed counter-attack led by Nicolo Barella, who raced into the opposition half, rode a challenge, and picked out Martinez for his cushioned volley.
Italy midfielder Barella became the first Inter player to provide assists in each of his first three matches of the Serie A season since Opta started collecting data in 2004-05.
But the champions again surrendered their advantage two minutes into the second half, when Augello was left unmarked to fire a volley into the top corner.