Tonev made a series of probing runs during the opening 30 minutes, putting centre-backs Skriniar and Miranda on the back foot, but a lack of support led to no final product from the Bulgaria global.
"We struggled a bit, Handanovic made two important saves but to win like that is a sign of adaptability, mental toughness and great personality", added the shaven-headed coach.
Inter, who came into the game on the back of three consecutive Serie A wins, were well shackled in the first half by their less glamorous counterparts, who kept the likes of Perisic, Mauro Icardi and Antonio Candreva relatively quiet during the opening exchanges.
"We are not interested in the table right now, we just did what we had to do".
Soccer Football - Serie A - FC Crotone vs Inter Milan - Stadio Ezio Scida, Crotone, Italy - September 16, 2017.
Joao Mario nearly gave Inter the lead in the first half, while Crotone's best chance went to Marcus Rohden, whose close-range header was superbly kept out by goalkeeper Inter keeper Samir Handanovic.
- Milan Skriniar has scored his first goal in Serie A, in his 42th appearance.
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"It was not easy and our players were very professional, adapting to a style of game they were unaccustomed to", Spalletti told Mediaset Premium.
Roma, last season's runners-up, have six points but have played only three games.
Edin Dzeko, who made headlines for criticising Di Francesco's tactics in the Atletico stalemate, hit back with a brace on 34 and 61 minutes with Alessandro Florenzi marking his comeback after almost a year out injured with an assist.
Inter's attacking trio failed to click into top gear, with Icardi a peripheral figure throughout, as Crotone finished the stronger against the disappointing Nerazzurri.
Samuel Souprayen saw red for Verona, who suffered their third loss in their return to the top flight, and are second from bottom.
Fiorentina leapfrogged mid-table Bologna in the early-season standings courtesy of a 2-1 derby dell'Appennino triumph.
Parity was restored within 60 seconds by Rodrigo Palacio, but German Pezzela had the decisive say with a looping header from a corner on 69 minutes.