Three second-half strikes saw Udinese rally from a halfime deficit for a 3-2 win over Liverpool that gave it a vital three points in Thursday night's Europa League Group A match.
However, Antonio Di Natale leveled the score early in the second half and a Sebastian Coates' own goal and Giovanni Pasquale's effort made it 3-1 before substitute Luis Suarez concluded a burst of three goals in five minutes when he reduced the deficit.
"I thought we were just lazy -- that is the easiest way to describe it," Rodgers said. "Just lazy in our play and loose and sloppy. It is as simple as that."
Liverpool made a bright start to the contest, with Oussama Assaidi shooting wide, Shelvey seeing an effort deflect off target and from the resulting corner, the unmarked Coates sending a header toward goal, which Zeljko Brkic just managed to keep out.
Udinese fired a warning at the other end when Jose Reina punched the ball in the direction of Di Natale, who responded with a strike that the Spaniard pushed away.
Reina then pulled off a superb stop to keep the score even, getting down to turn Medhi Benatia's header from a Di Natale free kick around the post.
The host went back on the attack and after Fabio Borini had dragged a shot across the face of goal having squirmed past a clutch of players, Brendan Rodgers' men made the breakthrough.
Shelvey collected the ball in midfield, sent it out wide to Downing, charged into the box and met the return cross with a crashing header past Brkic.
It was a third goal of the season for Shelvey, who had come off the bench to score a brace, which secured Liverpool a 5-3 victory in its opening group game against Young Boys.
The Reds looked to have things under control, but Di Natale went close again just before the half-hour mark with an effort that just cleared the bar.
Liverpool pressured once more but to no avail, with Glen Johnson firing over after Downing's shot was blocked, Jordan Henderson unable to make the most of a header and a Shelvey strike deflecting wide.
Rodgers had spoken in the build-up to the game about the need to keep tabs on Di Natale and less than a minute into the second half the 34-year-old punished the host for giving possession away.
Johnson lost the ball and it came to Di Natale, whose one-two with substitute Andrea Lazzari ended with the 34-year-old skipper expertly guiding a shot past Reina.
Rodgers reacted by introducing Steven Gerrard and Suarez, and the latter was unlucky not to restore Liverpool's lead when his acrobatic volley was inadvertently prevented from going in by the legs of his team-mate Shelvey.
Moments later, though, the Italians were in front themselves as Coates nodded a free kick into his own net with 20 minutes remaining.
Within two minutes the Italian side had scored again to leave Anfield stunned as Di Natale brought down a long ball and rolled it to Pasquale, who drilled in from outside the box.
The goal rush did not let up, with Suarez swiftly pulling one back direct from a free-kick with a delightful curling shot.
A further effort from the Uruguayan was then clawed away by Brkic, who also pushed a low strike by Suarez to safety.
Another substitute, Raheem Sterling, also showed plenty of endeavour with a couple of late attempts, but the equaliser would not come and Liverpool find themselves third in the group after this defeat and Anzhi Makhachkala's defeat of Young Boys in tonight's other Group A contest.
"In that opening period (of the second half) we didn't do what I call the dirty bit," Rodgers said. "In the first half we earned the right (to be in front) -- we pressed the ball, got tight and then showed our quality with the ball. But at the beginning of the second half we didn't do that, and before we knew it, we were 3-1 behind."
Information from Press Association was used in this report.