2019 we beat Dallas 12-10 on Sunday night and one of our field goals came from a Vonn Bell fumble recovery in DAL territory that on replay looked 50/50 on whether or not the runner was down by contact. It was exactly the kind of call a league could plausibly have go against a team they disliked without much criticism from anywhere but Saints fans.
The 2018 regular season game (reffed by none other than Bill @#$ing Vinovich) had a crucial moment where the Rams faked a punt in their own territory (Justin Hardee would later forget that the Rams loved faking punts in the NFCCG). Punter went out and was marked just shy. On the radio, our announcers said it looked like he got the 1st and it would be overturned. It did not and we ended up getting points from the field position. Before the NFCCG, Rams fans were legit mad Vinovich had been assigned to the game (sad, bitter laughter).
The 49ers were hopping mad in 2013 about some of the unnecessary roughness calls that went against them when sacking/hitting Brees. The Steelers were furious about a DPI call in the endzone (that I thought was DPI but I'll admit it was kinda ticky tack) that led to Saints points in 2018.
We've been at the receiving end of some awful officiating, no doubt. That Vikings game last year was definitely bad, I was there for it (in a way, i'm glad b/c I didn't see how bad the hands to the face call was until after I got back to NOLA, I'd have been way angrier if I'd watched that on TV). But you asked for examples where won close games where some close or questionable calls went our way, and those were the ones that leaped to mind.