The Three Lions never looked like slipping up in their latest 2014 World Cup qualifier, with skipper Steven Gerrard firing them in front after just 12 minutes.
Rickie Lambert nodded in a second before Welbeck completed the scoring with two strikes either side of the interval.
His two-goal salvo was, however, marred by a contentious caution which means the Manchester United forward must now sit out Tuesday's crucial trip to Ukraine.
Welbeck was booked for kicking the ball away moments after being flagged for offside.
Hodgson, who finds himself short on attacking options with Wayne Rooney unavailable and Daniel Sturridge nursing a knock, told Sky Sports: "My thoughts are really the same as everyone else's who have been involved in football, or followed football, or taken football seriously.
"It's almost unthinkable at that stage of the game, having made a 40-yard sprint and with maybe nothing more than a second between the ball arriving and striking it...It's not offside either. We've seen it and it wasn't even offside.
"To get the only yellow card of the game for that, and for that to then compound the yellow card he got in Montenegro - where he was fouled and we should have got a penalty, the referee didn't give it and gave him a yellow card instead - it doesn't get much more unfortunate than that.
"I think we're entitled to be aggrieved and I'm quite prepared to be bold and say that decision was, in my mind as a football person, totally incomprehensible."
On his depleted ranks, Hodgson added: "We said before these two games that we haven't been blessed with good fortune.
"To first lose Alex Oxlade-Chamberlain and then to lose Wayne Rooney and probably Daniel Sturridge, and now on top of that to lose Danny Welbeck, Glen Johnson and Phil Jones. Here we're talking about five players all of whom are very serious candidates for starting spots - some of them have never been outside of a starting spot.
"To go that wounded into such an important game, I can't say that's a positive thing and I can't make something so negative positive, but what I can say is that the group of players we have got and the ones that will play, they'll step up to the plate and they won't let us down.
"We can only hope that we won't be left regretting on Tuesday night the absence of some of these people."