Milan had been looking to close to within four point of Sinisa Mihajlovic's high-flyers and continue their late surge for a UEFA Europa League berth, but ultimately they fell short.
The hosts controlled the first half and saw numerous chances go begging, with Marco van Ginkel and Alessio Cerci going closest.
They were made to rue those missed opportunities as Roberto Soriano put the visitors ahead with just under an hour played.
Nigel de Jong levelled 16 minutes from the end as his bicycle kick went in off Alfred Duncan, but their late pressure proved fruitless and, although Inzaghi was pleased with the performance, he called for an improvement in front of goal.
He told Sky Sport Italia: "We played well and I am very happy with the performance. We had a great first half and forced Sampdoria to change their system.
"It's just a shame we created so many crosses and passes into the area without getting a goal. It was one of our best halves of the season.
"Sampdoria changed around and did better for 20 minutes, scoring the goal. But we fought back and after the equaliser hit the post with Suso.
"We had 14 shots, seven of them were on target and we had about 60 per cent possession. We did all we needed to, except being more decisive in front of goal.
"I can't ask for any more from my players aside from precision in front of goal. We certainly should have done more this season overall, but I think we gave more or less everything we had in this game."
Get the best features, fun and footballing frolics straight to your inbox every week.
Thank you for signing up to Four Four Two. You will receive a verification email shortly.
There was a problem. Please refresh the page and try again.