Patel and Kartik featured in a 95-run partnership for the unbroken seventh wicket stand as the Rest plundered 114 runs in the post-tea session to wrest back the initiative from the Ranji Trophy champions at the PCA Stadium.
Patel remained unbeaten on 56 while Kartik was 45 not out, having faced just 69 balls and hit six boundaries at stumps.
The duo's brilliant counter attack that denied Mumbai the advantage of second new ball not only made up for the Rest's top order collapse, which saw them slump to 198 for six soon after tea, but also overshadowed Sridharan Sriram's painstaking half-century that actually began the rebuilding process.
Sriram's 77 consumed 196 balls, spanned 261 minutes and contained seven fours. The Tamil Nadu left-hander also was associated in a 60-run partnership with Venugopala Rao (20) for the fourth wicket.
Earlier, Mumbai were given a brilliant start by their seamers, particularly Avishkar Salvi. The right-arm pacer finished with the lowest count, 1-70, among the Mumbai bowlers but was clearly the most impressive.
Sairaj Bahutule scalped three for 50 and Robin Morris picked two for 57 but it was Salvi's disconcerting movement and bounce that rattled the Rest batsmen.
Even when the second new ball failed to fetch any wicket in the evening and the batsmen found stroke making easy as the pitch dried up, Salvi had them hopping.
Munaf Patel proved to be the lone disappointment for coach Chandrakanth Pandit as he sprayed the new ball and failed to use the early juice in the wicket.
Patel's sorry day at office was complete when, in the 10th over in his second spell, he fell on his follow through and hurt his shoulder.
Pandit said he decided against continuing with him as he was in severe pain. An MRI scan report on the injury was being awaited.
But the former India stumper was furious with the way his players let the opposition off the hook. "It was a terrible performance, we gave away whatever advantage we had," he said.
"We had a great opportunity at 198 for six but we gave away more than 100 runs in the last session."
He defended the team's decision to bowl on a hard wicket with a tinge of grass.
"The conditions were not easy in the morning. Remember we had them six down early in the day," he said.
"That is what sport is all about, it is about handling the important situations. They did not handle well. Patel and Kartik played their shots later in the day."
Salvi struck in the seventh when he dived forward on his follow through to collect a low leading edge from Aakash Chopra.
Goutam Gambhir was fearless in playing his strokes but after racing to 39 from 44 balls with seven fours, he was done in by the bounce of Morris and edged a simple catch to Vinayak Mane in the slip cordon.