Argentina had started their ultimately successful 2022 campaign with defeat to Saudi Arabia, but avoided any such upset on this occasion thanks to Messi's first World Cup treble on Tuesday (Wednesday AEST).
Messi's second took him level with Brazil great Ronaldo on 15 goals across his six World Cup appearances.
He was denied in a one-on-one by Algeria goalkeeper Luca Zidane and had a penalty appeal turned down before joining Germany's Klose in a share of the record.
"When I'm in good shape, I give it my all," Messi, who had been hampered by a hamstring problem in the lead-up to the tournament, said through an interpreter.
"It makes me very happy to have lived through everything that came my way. What I'm living though now is the cherry on top.
"I'm very happy an grateful for this wonderful group. I enjoy it so much."
The 38-year-old will have chances to move into the outright lead in Argentina's remaining Group J games against Austria and Jordan next week – assuming France's Kylian Mbappe, who is on 14 after a brace against Senegal, does not run riot against Iraq on Monday.
Messi thought he had opened the scoring after only five minutes, but was denied by a tight offside call after a composed finish into the bottom corner.
Four minutes later it was Algeria's turn to be denied – this time after VAR overruled the original decision when Ibrahim Maza played in Fares Chaibi and he put a low effort past Emi Martinez.
It proved to be the only attempt on target a toothless Algeria side – which surprisingly saw Riyad Mahrez dropped to the bench – could muster all evening.
Messi opened the scoring after 17 minutes as he linked up with Inter Miami teammate Rodrigo de Paul and fired a 20-yard effort into the roof of the net.
Chaibi wasted an excellent chance to equalise and, at the other end, Thiago Almada fired over before the break.
Lautaro Martinez shot straight at Algeria goalkeeper Zidane – the son of France World Cup winner Zinedine – who also saved well from Alexis Mac Allister on the hour mark, only for Messi to react quickest and turn in the rebound.
Mac Allister's defence-splitting pass sent Messi clear after 66 minutes, but Zidane was equal to his effort and referee Szymon Marciniak waved away the Inter Miami forward's penalty claims after Manchester City defender Rayan Ait-Nouri sent him tumbling in the box.
Messi's moment did finally arrive in the 76th minute as he collected Nicolas Gonzalez's pass and fired a low effort into the bottom corner.
Messi made way for Nico Paz a short time afterwards to a standing ovation from the Arrowhead Stadium crowd as Argentina comfortably saw out an opening win.
"At a loss for words about Leo. What can I say? He's incredible," Argentina coach Lionel Scaloni said.
"Everyone in football wants to see him and enjoys it."
With AP.
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